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Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubccocl 15701 A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ph  ->  J  e.  (Subcat `  C )
 )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  Y  e.  S )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  F  e.  ( X J Y ) )   &    |-  ( ph  ->  G  e.  ( Y J Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  e.  ( X J Z ) )
 
Theoremsubccatid 15702* A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  .1.  =  ( Id `  C )   =>    |-  ( ph  ->  ( D  e.  Cat  /\  ( Id `  D )  =  ( x  e.  S  |->  (  .1.  `  x )
 ) ) )
 
Theoremsubcid 15703 The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  X  e.  S )   =>    |-  ( ph  ->  (  .1.  `  X )  =  ( ( Id `  D ) `  X ) )
 
Theoremsubccat 15704 A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  J  e.  (Subcat `  C ) )   =>    |-  ( ph  ->  D  e.  Cat )
 
Theoremissubc3 15705* Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 16546, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  H  =  ( Hom f  `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  D  =  ( C  |`cat  J )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  J  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  H  /\  A. x  e.  S  (  .1.  `  x )  e.  ( x J x )  /\  D  e.  Cat ) ) )
 
Theoremfullsubc 15706 The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory), see definition 4.1(2) of [Adamek] p. 48. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom f  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  C_  B )   =>    |-  ( ph  ->  ( H  |`  ( S  X.  S ) )  e.  (Subcat `  C )
 )
 
Theoremfullresc 15707 The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom f  `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  S  C_  B )   &    |-  D  =  ( Cs  S )   &    |-  E  =  ( C  |`cat  ( H  |`  ( S  X.  S ) ) )   =>    |-  ( ph  ->  (
 ( Hom f  `  D )  =  ( Hom f  `  E )  /\  (compf `  D )  =  (compf `  E ) ) )
 
Theoremresscat 15708 A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ( C  e.  Cat  /\  S  e.  V ) 
 ->  ( Cs  S )  e.  Cat )
 
Theoremsubsubc 15709 A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  D  =  ( C  |`cat  H )   =>    |-  ( H  e.  (Subcat `  C )  ->  ( J  e.  (Subcat `  D ) 
 <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H ) ) )
 
8.1.7  Functors
 
Syntaxcfunc 15710 Extend class notation with the class of all functors.
 class  Func
 
Syntaxcidfu 15711 Extend class notation with identity functor.
 class idfunc
 
Syntaxccofu 15712 Extend class notation with functor composition.
 class  o.func
 
Syntaxcresf 15713 Extend class notation to include restriction of a functor to a subcategory.
 class  |`f
 
Definitiondf-func 15714* Function returning all the functors from a category  t to a category  u. Definition 3.17 of [Adamek] p. 29, and definition in [Lang] p. 62 ("covariant functor"). Intuitively a functor associates any morphism of  t to a morphism of  u, any object of  t to an object of  u, and respects the identity, the composition, the domain and the codomain. Here to capture the idea that a functor associates any object of  t to an object of  u we write it associates any identity of  t to an identity of  u which simplifies the definition. According to remark 3.19 in [Adamek] p. 30, "a functor F : A -> B is technically a family of functions; one from Ob(A) to Ob(B) [here: f, called "the object part" in the following], and for each pair (A,A') of A-objects, one from hom(A,A') to hom(FA, FA') [here: g, called "the morphism part" in the following]". (Contributed by FL, 10-Feb-2008.) (Revised by Mario Carneiro, 2-Jan-2017.)
 |- 
 Func  =  ( t  e.  Cat ,  u  e. 
 Cat  |->  { <. f ,  g >.  |  [. ( Base `  t )  /  b ]. ( f : b --> ( Base `  u )  /\  g  e.  X_ z  e.  ( b  X.  b
 ) ( ( ( f `  ( 1st `  z ) ) ( Hom  `  u )
 ( f `  ( 2nd `  z ) ) )  ^m  ( ( Hom  `  t ) `  z ) )  /\  A. x  e.  b  ( ( ( x g x ) `  (
 ( Id `  t
 ) `  x )
 )  =  ( ( Id `  u ) `
  ( f `  x ) )  /\  A. y  e.  b  A. z  e.  b  A. m  e.  ( x ( Hom  `  t )
 y ) A. n  e.  ( y ( Hom  `  t ) z ) ( ( x g z ) `  ( n ( <. x ,  y >. (comp `  t
 ) z ) m ) )  =  ( ( ( y g z ) `  n ) ( <. ( f `
  x ) ,  ( f `  y
 ) >. (comp `  u ) ( f `  z ) ) ( ( x g y ) `  m ) ) ) ) }
 )
 
Definitiondf-idfu 15715* Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- idfunc  =  ( t  e.  Cat  |->  [_ ( Base `  t )  /  b ]_ <. (  _I  |`  b ) ,  (
 z  e.  ( b  X.  b )  |->  (  _I  |`  ( ( Hom  `  t ) `  z ) ) )
 >. )
 
Definitiondf-cofu 15716* Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- 
 o.func  =  ( g  e.  _V ,  f  e.  _V  |->  <.
 ( ( 1st `  g
 )  o.  ( 1st `  f ) ) ,  ( x  e.  dom  dom  ( 2nd `  f
 ) ,  y  e. 
 dom  dom  ( 2nd `  f
 )  |->  ( ( ( ( 1st `  f
 ) `  x )
 ( 2nd `  g )
 ( ( 1st `  f
 ) `  y )
 )  o.  ( x ( 2nd `  f
 ) y ) ) ) >. )
 
Definitiondf-resf 15717* Define the restriction of a functor to a subcategory (analogue of df-res 4866). (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  |`f  =  ( f  e.  _V ,  h  e.  _V  |->  <.
 ( ( 1st `  f
 )  |`  dom  dom  h ) ,  ( x  e. 
 dom  h  |->  ( ( ( 2nd `  f
 ) `  x )  |`  ( h `  x ) ) ) >. )
 
Theoremrelfunc 15718 The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |- 
 Rel  ( D  Func  E )
 
Theoremfuncrcl 15719 Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( F  e.  ( D  Func  E )  ->  ( D  e.  Cat  /\  E  e.  Cat )
 )
 
Theoremisfunc 15720* Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  C  =  (
 Base `  E )   &    |-  H  =  ( Hom  `  D )   &    |-  J  =  ( Hom  `  E )   &    |-  .1.  =  ( Id `  D )   &    |-  I  =  ( Id `  E )   &    |-  .x.  =  (comp `  D )   &    |-  O  =  (comp `  E )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   =>    |-  ( ph  ->  ( F ( D  Func  E ) G  <->  ( F : B
 --> C  /\  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z )
 ) J ( F `
  ( 2nd `  z
 ) ) )  ^m  ( H `  z ) )  /\  A. x  e.  B  ( ( ( x G x ) `
  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x H y ) A. n  e.  ( y H z ) ( ( x G z ) `  ( n ( <. x ,  y >.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) ) ) ) ) )
 
Theoremisfuncd 15721* Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  C  =  (
 Base `  E )   &    |-  H  =  ( Hom  `  D )   &    |-  J  =  ( Hom  `  E )   &    |-  .1.  =  ( Id `  D )   &    |-  I  =  ( Id `  E )   &    |-  .x.  =  (comp `  D )   &    |-  O  =  (comp `  E )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  E  e.  Cat )   &    |-  ( ph  ->  F : B --> C )   &    |-  ( ph  ->  G  Fn  ( B  X.  B ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x G y ) : ( x H y ) --> ( ( F `
  x ) J ( F `  y
 ) ) )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (
 ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  /\  ( m  e.  ( x H y )  /\  n  e.  ( y H z ) ) )  ->  ( ( x G z ) `  ( n ( <. x ,  y >.  .x.  z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y ) >. O ( F `  z ) ) ( ( x G y ) `  m ) ) )   =>    |-  ( ph  ->  F ( D  Func  E ) G )
 
Theoremfuncf1 15722 The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  C  =  (
 Base `  E )   &    |-  ( ph  ->  F ( D 
 Func  E ) G )   =>    |-  ( ph  ->  F : B
 --> C )
 
Theoremfuncixp 15723* The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  H  =  ( Hom  `  D )   &    |-  J  =  ( Hom  `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   =>    |-  ( ph  ->  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z )
 ) J ( F `
  ( 2nd `  z
 ) ) )  ^m  ( H `  z ) ) )
 
Theoremfuncf2 15724 The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  H  =  ( Hom  `  D )   &    |-  J  =  ( Hom  `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X G Y ) : ( X H Y )
 --> ( ( F `  X ) J ( F `  Y ) ) )
 
Theoremfuncfn2 15725 The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  ( ph  ->  F ( D  Func  E ) G )   =>    |-  ( ph  ->  G  Fn  ( B  X.  B ) )
 
Theoremfuncid 15726 A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  .1.  =  ( Id `  D )   &    |-  I  =  ( Id `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( ( X G X ) `  (  .1.  `  X ) )  =  ( I `  ( F `  X ) ) )
 
Theoremfuncco 15727 A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  H  =  ( Hom  `  D )   &    |-  .x.  =  (comp `  D )   &    |-  O  =  (comp `  E )   &    |-  ( ph  ->  F ( D 
 Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  M  e.  ( X H Y ) )   &    |-  ( ph  ->  N  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( ( X G Z ) `  ( N (
 <. X ,  Y >.  .x. 
 Z ) M ) )  =  ( ( ( Y G Z ) `  N ) (
 <. ( F `  X ) ,  ( F `  Y ) >. O ( F `  Z ) ) ( ( X G Y ) `  M ) ) )
 
Theoremfuncsect 15728 The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  S  =  (Sect `  D )   &    |-  T  =  (Sect `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M ( X S Y ) N )   =>    |-  ( ph  ->  (
 ( X G Y ) `  M ) ( ( F `  X ) T ( F `  Y ) ) ( ( Y G X ) `  N ) )
 
Theoremfuncinv 15729 The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  I  =  (Inv `  D )   &    |-  J  =  (Inv `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M ( X I Y ) N )   =>    |-  ( ph  ->  (
 ( X G Y ) `  M ) ( ( F `  X ) J ( F `  Y ) ) ( ( Y G X ) `  N ) )
 
Theoremfunciso 15730 The image of an isomorphism under a functor is an isomorphism. Proposition 3.21 of [Adamek] p. 32. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  D )   &    |-  I  =  ( 
 Iso  `  D )   &    |-  J  =  (  Iso  `  E )   &    |-  ( ph  ->  F ( D  Func  E ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M  e.  ( X I Y ) )   =>    |-  ( ph  ->  (
 ( X G Y ) `  M )  e.  ( ( F `  X ) J ( F `  Y ) ) )
 
Theoremfuncoppc 15731 A functor on categories yields a functor on the opposite categories (in the same direction), see definition 3.41 of [Adamek] p. 39. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  P  =  (oppCat `  D )   &    |-  ( ph  ->  F ( C  Func  D ) G )   =>    |-  ( ph  ->  F ( O  Func  P )tpos 
 G )
 
Theoremidfuval 15732* Value of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `
  z ) ) ) >. )
 
Theoremidfu2nd 15733 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X ( 2nd `  I
 ) Y )  =  (  _I  |`  ( X H Y ) ) )
 
Theoremidfu2 15734 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( ( X ( 2nd `  I
 ) Y ) `  F )  =  F )
 
Theoremidfu1st 15735 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   =>    |-  ( ph  ->  ( 1st `  I )  =  (  _I  |`  B ) )
 
Theoremidfu1 15736 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( 1st `  I ) `  X )  =  X )
 
Theoremidfucl 15737 The identity functor is a functor. Example 3.20(1) of [Adamek] p. 30. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   =>    |-  ( C  e.  Cat  ->  I  e.  ( C  Func  C ) )
 
Theoremcofuval 15738* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   =>    |-  ( ph  ->  ( G  o.func 
 F )  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `  x ) ( 2nd `  G )
 ( ( 1st `  F ) `  y ) )  o.  ( x ( 2nd `  F )
 y ) ) )
 >. )
 
Theoremcofu1st 15739 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   =>    |-  ( ph  ->  ( 1st `  ( G  o.func  F ) )  =  ( ( 1st `  G )  o.  ( 1st `  F ) ) )
 
Theoremcofu1 15740 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( ( 1st `  ( G  o.func 
 F ) ) `  X )  =  (
 ( 1st `  G ) `  ( ( 1st `  F ) `  X ) ) )
 
Theoremcofu2nd 15741 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X ( 2nd `  ( G  o.func 
 F ) ) Y )  =  ( ( ( ( 1st `  F ) `  X ) ( 2nd `  G )
 ( ( 1st `  F ) `  Y ) )  o.  ( X ( 2nd `  F ) Y ) ) )
 
Theoremcofu2 15742 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( ( X ( 2nd `  ( G  o.func  F )
 ) Y ) `  R )  =  (
 ( ( ( 1st `  F ) `  X ) ( 2nd `  G ) ( ( 1st `  F ) `  Y ) ) `  (
 ( X ( 2nd `  F ) Y ) `
  R ) ) )
 
Theoremcofuval2 15743* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  ( ph  ->  F ( C  Func  D ) G )   &    |-  ( ph  ->  H ( D  Func  E ) K )   =>    |-  ( ph  ->  ( <. H ,  K >.  o.func  <. F ,  G >. )  = 
 <. ( H  o.  F ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( F `  x ) K ( F `  y ) )  o.  ( x G y ) ) ) >. )
 
Theoremcofucl 15744 The composition of two functors is a functor. Proposition 3.23 of [Adamek] p. 33. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( D  Func  E ) )   =>    |-  ( ph  ->  ( G  o.func 
 F )  e.  ( C  Func  E ) )
 
Theoremcofuass 15745 Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  ( ph  ->  H  e.  ( D  Func  E ) )   &    |-  ( ph  ->  K  e.  ( E  Func  F ) )   =>    |-  ( ph  ->  (
 ( K  o.func  H )  o.func  G )  =  ( K  o.func  ( H  o.func  G )
 ) )
 
Theoremcofulid 15746 The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  I  =  (idfunc `  D )   =>    |-  ( ph  ->  ( I  o.func 
 F )  =  F )
 
Theoremcofurid 15747 The identity functor is a right identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  I  =  (idfunc `  C )   =>    |-  ( ph  ->  ( F  o.func 
 I )  =  F )
 
Theoremresfval 15748* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  H  e.  W )   =>    |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F )  |`  dom  dom  H ) ,  ( x  e. 
 dom  H  |->  ( ( ( 2nd `  F ) `  x )  |`  ( H `  x ) ) ) >. )
 
Theoremresfval2 15749* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  H  e.  W )   &    |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( <. F ,  G >.  |`f  H )  =  <. ( F  |`  S ) ,  ( x  e.  S ,  y  e.  S  |->  ( ( x G y )  |`  ( x H y ) ) ) >. )
 
Theoremresf1st 15750 Value of the functor restriction operator on objects. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  H  e.  W )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   =>    |-  ( ph  ->  ( 1st `  ( F  |`f  H ) )  =  ( ( 1st `  F )  |`  S ) )
 
Theoremresf2nd 15751 Value of the functor restriction operator on morphisms. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  H  e.  W )   &    |-  ( ph  ->  H  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   =>    |-  ( ph  ->  ( X ( 2nd `  ( F  |`f  H ) ) Y )  =  ( ( X ( 2nd `  F ) Y )  |`  ( X H Y ) ) )
 
Theoremfuncres 15752 A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  H  e.  (Subcat `  C ) )   =>    |-  ( ph  ->  ( F  |`f  H )  e.  (
 ( C  |`cat  H )  Func  D ) )
 
Theoremfuncres2b 15753* Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  A  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  R  e.  (Subcat `  D ) )   &    |-  ( ph  ->  R  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  F : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A ) )  ->  ( x G y ) : Y --> ( ( F `
  x ) R ( F `  y
 ) ) )   =>    |-  ( ph  ->  ( F ( C  Func  D ) G  <->  F ( C  Func  ( D  |`cat  R ) ) G ) )
 
Theoremfuncres2 15754 A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  ( R  e.  (Subcat `  D )  ->  ( C  Func  ( D  |`cat  R )
 )  C_  ( C  Func  D ) )
 
Theoremwunfunc 15755 A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  C  e.  U )   &    |-  ( ph  ->  D  e.  U )   =>    |-  ( ph  ->  ( C  Func  D )  e.  U )
 
Theoremfuncpropd 15756 If two categories have the same set of objects, morphisms, and compositions, then they have the same functors. (Contributed by Mario Carneiro, 17-Jan-2017.)
 |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  ( A  Func  C )  =  ( B  Func  D ) )
 
Theoremfuncres2c 15757 Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
 |-  A  =  ( Base `  C )   &    |-  E  =  ( Ds  S )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  F : A --> S )   =>    |-  ( ph  ->  ( F ( C  Func  D ) G  <->  F ( C  Func  E ) G ) )
 
8.1.8  Full & faithful functors
 
Syntaxcful 15758 Extend class notation with the class of all full functors.
 class Full
 
Syntaxcfth 15759 Extend class notation with the class of all faithful functors.
 class Faith
 
Definitiondf-full 15760* Function returning all the full functors from a category  C to a category  D. A full functor is a functor in which all the morphism maps  G ( X ,  Y ) between objects  X ,  Y  e.  C are surjections. Definition 3.27(3) in [Adamek] p. 34. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |- Full  =  ( c  e.  Cat ,  d  e.  Cat  |->  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c ) ran  ( x g y )  =  ( ( f `  x ) ( Hom  `  d
 ) ( f `  y ) ) ) } )
 
Definitiondf-fth 15761* Function returning all the faithful functors from a category  C to a category  D. A full functor is a functor in which all the morphism maps  G ( X ,  Y ) between objects  X ,  Y  e.  C are injections. Definition 3.27(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |- Faith  =  ( c  e.  Cat ,  d  e.  Cat  |->  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c ) Fun  `' ( x g y ) ) } )
 
Theoremfullfunc 15762 A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( C Full  D ) 
 C_  ( C  Func  D )
 
Theoremfthfunc 15763 A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( C Faith  D ) 
 C_  ( C  Func  D )
 
Theoremrelfull 15764 The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |- 
 Rel  ( C Full  D )
 
Theoremrelfth 15765 The set of faithful functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |- 
 Rel  ( C Faith  D )
 
Theoremisfull 15766* Value of the set of full functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  J  =  ( Hom  `  D )   =>    |-  ( F ( C Full  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ran  ( x G y )  =  ( ( F `  x ) J ( F `  y ) ) ) )
 
Theoremisfull2 15767* Equivalent condition for a full functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  J  =  ( Hom  `  D )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( F ( C Full 
 D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y )
 -onto-> ( ( F `  x ) J ( F `  y ) ) ) )
 
Theoremfullfo 15768 The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  J  =  ( Hom  `  D )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  F ( C Full  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X G Y ) : ( X H Y ) -onto-> ( ( F `
  X ) J ( F `  Y ) ) )
 
Theoremfulli 15769* The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  J  =  ( Hom  `  D )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  F ( C Full  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( ( F `  X ) J ( F `  Y ) ) )   =>    |-  ( ph  ->  E. f  e.  ( X H Y ) R  =  (
 ( X G Y ) `  f ) )
 
Theoremisfth 15770* Value of the set of faithful functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   =>    |-  ( F ( C Faith  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  Fun  `' ( x G y ) ) )
 
Theoremisfth2 15771* Equivalent condition for a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  J  =  ( Hom  `  D )   =>    |-  ( F ( C Faith  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y )
 -1-1-> ( ( F `  x ) J ( F `  y ) ) ) )
 
Theoremisffth2 15772* A fully faithful functor is a functor which is bijective on hom-sets. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  J  =  ( Hom  `  D )   =>    |-  ( F ( ( C Full  D )  i^i  ( C Faith  D ) ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y ) -1-1-onto-> ( ( F `  x ) J ( F `  y ) ) ) )
 
Theoremfthf1 15773 The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  J  =  ( Hom  `  D )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-> ( ( F `
  X ) J ( F `  Y ) ) )
 
Theoremfthi 15774 The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  J  =  ( Hom  `  D )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   &    |-  ( ph  ->  S  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( ( ( X G Y ) `  R )  =  ( ( X G Y ) `  S )  <->  R  =  S ) )
 
Theoremffthf1o 15775 The morphism map of a fully faithful functor is a bijection. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  J  =  ( Hom  `  D )   &    |-  ( ph  ->  F ( ( C Full  D )  i^i  ( C Faith  D ) ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-onto-> ( ( F `  X ) J ( F `  Y ) ) )
 
Theoremfullpropd 15776 If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  ( A Full  C )  =  ( B Full  D ) )
 
Theoremfthpropd 15777 If two categories have the same set of objects, morphisms, and compositions, then they have the same faithful functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  ( A Faith  C )  =  ( B Faith  D ) )
 
Theoremfulloppc 15778 The opposite functor of a full functor is also full. Proposition 3.43(d) in [Adamek] p. 39. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  P  =  (oppCat `  D )   &    |-  ( ph  ->  F ( C Full  D ) G )   =>    |-  ( ph  ->  F ( O Full  P )tpos  G )
 
Theoremfthoppc 15779 The opposite functor of a faithful functor is also faithful. Proposition 3.43(c) in [Adamek] p. 39. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  P  =  (oppCat `  D )   &    |-  ( ph  ->  F ( C Faith  D ) G )   =>    |-  ( ph  ->  F ( O Faith  P )tpos  G )
 
Theoremffthoppc 15780 The opposite functor of a fully faithful functor is also full and faithful. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  O  =  (oppCat `  C )   &    |-  P  =  (oppCat `  D )   &    |-  ( ph  ->  F ( ( C Full  D )  i^i  ( C Faith  D ) ) G )   =>    |-  ( ph  ->  F (
 ( O Full  P )  i^i  ( O Faith  P ) )tpos  G )
 
Theoremfthsect 15781 A faithful functor reflects sections. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M  e.  ( X H Y ) )   &    |-  ( ph  ->  N  e.  ( Y H X ) )   &    |-  S  =  (Sect `  C )   &    |-  T  =  (Sect `  D )   =>    |-  ( ph  ->  ( M ( X S Y ) N  <->  ( ( X G Y ) `  M ) ( ( F `  X ) T ( F `  Y ) ) ( ( Y G X ) `  N ) ) )
 
Theoremfthinv 15782 A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  M  e.  ( X H Y ) )   &    |-  ( ph  ->  N  e.  ( Y H X ) )   &    |-  I  =  (Inv `  C )   &    |-  J  =  (Inv `  D )   =>    |-  ( ph  ->  ( M ( X I Y ) N  <->  ( ( X G Y ) `  M ) ( ( F `  X ) J ( F `  Y ) ) ( ( Y G X ) `  N ) ) )
 
Theoremfthmon 15783 A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   &    |-  M  =  (Mono `  C )   &    |-  N  =  (Mono `  D )   &    |-  ( ph  ->  ( ( X G Y ) `  R )  e.  ( ( F `  X ) N ( F `  Y ) ) )   =>    |-  ( ph  ->  R  e.  ( X M Y ) )
 
Theoremfthepi 15784 A faithful functor reflects epimorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   &    |-  E  =  (Epi `  C )   &    |-  P  =  (Epi `  D )   &    |-  ( ph  ->  ( ( X G Y ) `  R )  e.  ( ( F `  X ) P ( F `  Y ) ) )   =>    |-  ( ph  ->  R  e.  ( X E Y ) )
 
Theoremffthiso 15785 A fully faithful functor reflects isomorphisms. Corollary 3.32 of [Adamek] p. 35. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  B  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  F ( C Faith  D ) G )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   &    |-  ( ph  ->  F ( C Full  D ) G )   &    |-  I  =  (  Iso  `  C )   &    |-  J  =  ( 
 Iso  `  D )   =>    |-  ( ph  ->  ( R  e.  ( X I Y )  <-> 
 ( ( X G Y ) `  R )  e.  ( ( F `  X ) J ( F `  Y ) ) ) )
 
Theoremfthres2b 15786* Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  A  =  ( Base `  C )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  R  e.  (Subcat `  D ) )   &    |-  ( ph  ->  R  Fn  ( S  X.  S ) )   &    |-  ( ph  ->  F : A
 --> S )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A ) )  ->  ( x G y ) : Y --> ( ( F `
  x ) R ( F `  y
 ) ) )   =>    |-  ( ph  ->  ( F ( C Faith  D ) G  <->  F ( C Faith  ( D  |`cat  R ) ) G ) )
 
Theoremfthres2c 15787 Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
 |-  A  =  ( Base `  C )   &    |-  E  =  ( Ds  S )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  F : A --> S )   =>    |-  ( ph  ->  ( F ( C Faith  D ) G  <->  F ( C Faith  E ) G ) )
 
Theoremfthres2 15788 A faithful functor into a restricted category is also a faithful functor into the whole category. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( R  e.  (Subcat `  D )  ->  ( C Faith  ( D  |`cat  R )
 )  C_  ( C Faith  D ) )
 
Theoremidffth 15789 The identity functor is a fully faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  I  =  (idfunc `  C )   =>    |-  ( C  e.  Cat  ->  I  e.  ( ( C Full  C )  i^i  ( C Faith  C ) ) )
 
Theoremcofull 15790 The composition of two full functors is full. Proposition 3.30(d) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  ( ph  ->  F  e.  ( C Full  D ) )   &    |-  ( ph  ->  G  e.  ( D Full  E ) )   =>    |-  ( ph  ->  ( G  o.func 
 F )  e.  ( C Full  E ) )
 
Theoremcofth 15791 The composition of two faithful functors is faithful. Proposition 3.30(c) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  ( ph  ->  F  e.  ( C Faith  D ) )   &    |-  ( ph  ->  G  e.  ( D Faith  E ) )   =>    |-  ( ph  ->  ( G  o.func 
 F )  e.  ( C Faith  E ) )
 
Theoremcoffth 15792 The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  ( ph  ->  F  e.  ( ( C Full  D )  i^i  ( C Faith  D ) ) )   &    |-  ( ph  ->  G  e.  (
 ( D Full  E )  i^i  ( D Faith  E ) ) )   =>    |-  ( ph  ->  ( G  o.func 
 F )  e.  (
 ( C Full  E )  i^i  ( C Faith  E ) ) )
 
Theoremrescfth 15793 The inclusion functor from a subcategory is a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  D  =  ( C  |`cat  J )   &    |-  I  =  (idfunc `  D )   =>    |-  ( J  e.  (Subcat `  C )  ->  I  e.  ( D Faith  C ) )
 
Theoremressffth 15794 The inclusion functor from a full subcategory is a full and faithful functor, see also remark 4.4(2) in [Adamek] p. 49. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  D  =  ( Cs  S )   &    |-  I  =  (idfunc `  D )   =>    |-  ( ( C  e.  Cat  /\  S  e.  V ) 
 ->  I  e.  (
 ( D Full  C )  i^i  ( D Faith  C ) ) )
 
Theoremfullres2c 15795 Condition for a full functor to also be a full functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
 |-  A  =  ( Base `  C )   &    |-  E  =  ( Ds  S )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  F : A --> S )   =>    |-  ( ph  ->  ( F ( C Full  D ) G  <->  F ( C Full  E ) G ) )
 
Theoremffthres2c 15796 Condition for a fully faithful functor to also be a fully faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  A  =  ( Base `  C )   &    |-  E  =  ( Ds  S )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  F : A --> S )   =>    |-  ( ph  ->  ( F ( ( C Full 
 D )  i^i  ( C Faith  D ) ) G  <->  F ( ( C Full 
 E )  i^i  ( C Faith  E ) ) G ) )
 
8.1.9  Natural transformations and the functor category
 
Syntaxcnat 15797 Extend class notation to include the collection of natural transformations.
 class Nat
 
Syntaxcfuc 15798 Extend class notation to include the functor category.
 class FuncCat
 
Definitiondf-nat 15799* Definition of a natural transformation between two functors. A natural transformation  A : F --> G is a collection of arrows  A ( x ) : F ( x ) --> G ( x ), such that  A ( y )  o.  F ( h )  =  G ( h )  o.  A ( x ) for each morphism  h : x --> y. Definition 6.1 in [Adamek] p. 83, and definition in [Lang] p. 65. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |- Nat 
 =  ( t  e. 
 Cat ,  u  e.  Cat  |->  ( f  e.  (
 t  Func  u ) ,  g  e.  ( t 
 Func  u )  |->  [_ ( 1st `  f )  /  r ]_ [_ ( 1st `  g )  /  s ]_ { a  e.  X_ x  e.  ( Base `  t ) ( ( r `  x ) ( Hom  `  u ) ( s `  x ) )  | 
 A. x  e.  ( Base `  t ) A. y  e.  ( Base `  t ) A. h  e.  ( x ( Hom  `  t ) y ) ( ( a `  y ) ( <. ( r `  x ) ,  ( r `  y ) >. (comp `  u ) ( s `
  y ) ) ( ( x ( 2nd `  f )
 y ) `  h ) )  =  (
 ( ( x ( 2nd `  g )
 y ) `  h ) ( <. ( r `
  x ) ,  ( s `  x ) >. (comp `  u ) ( s `  y ) ) ( a `  x ) ) } ) )
 
Definitiondf-fuc 15800* Definition of the category of functors between two fixed categories, with the objects being functors and the morphisms being natural transformations. Definition 6.15 in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |- FuncCat  =  ( t  e.  Cat ,  u  e.  Cat  |->  { <. (
 Base `  ndx ) ,  ( t  Func  u ) >. ,  <. ( Hom  `  ndx ) ,  (
 t Nat  u ) >. , 
 <. (comp `  ndx ) ,  ( v  e.  (
 ( t  Func  u )  X.  ( t  Func  u ) ) ,  h  e.  ( t  Func  u )  |->  [_ ( 1st `  v
 )  /  f ]_ [_ ( 2nd `  v
 )  /  g ]_ ( b  e.  (
 g ( t Nat  u ) h ) ,  a  e.  ( f ( t Nat 
 u ) g ) 
 |->  ( x  e.  ( Base `  t )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x ) ,  (
 ( 1st `  g ) `  x ) >. (comp `  u ) ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) ) )
 >. } )
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