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Theorem List for Metamath Proof Explorer - 13601-13700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcofu1st 13601 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 13602 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 13603 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 13604 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 13605* 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 13606 The composition of two functors is a functor. (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 13607 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 13608 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 13609 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 13610* 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 13611* 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 13612 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 13613 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 13614 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 13615* 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 13616 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 13617 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 13618 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  ->  (  Homf  `  A )  =  ( 
 Homf  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  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 13619 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 ) )
 
8.1.7  Full & faithful functors
 
Syntaxcful 13620 Extend class notation with the class of all full functors.
 class Full
 
Syntaxcfth 13621 Extend class notation with the class of all faithful functors.
 class Faith
 
Definitiondf-full 13622* 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. (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 13623* 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. (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 13624 A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( C Full  D ) 
 C_  ( C  Func  D )
 
Theoremfthfunc 13625 A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( C Faith  D ) 
 C_  ( C  Func  D )
 
Theoremrelfull 13626 The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |- 
 Rel  ( C Full  D )
 
Theoremrelfth 13627 The set of faithful functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |- 
 Rel  ( C Faith  D )
 
Theoremisfull 13628* 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 13629* 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 13630 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 13631* 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 13632* 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 13633* 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 13634* 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 13635 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 13636 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 13637 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 13638 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  ->  (  Homf  `  A )  =  ( 
 Homf  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  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 13639 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  ->  (  Homf  `  A )  =  ( 
 Homf  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  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 13640 The opposite functor of a full functor is also full. (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 13641 The opposite functor of a faithful functor is also faithful. (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 13642 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 13643 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 13644 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 13645 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 13646 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 13647 A fully faithful functor reflects isomorphisms. (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 13648* 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 13649 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 13650 A functor into a restricted category is also a 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 13651 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 13652 The composition of two full functors is full. (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 13653 The composition of two faithful functors is faithful. (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 13654 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 13655 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 13656 The inclusion functor from a full subcategory is a full and faithful functor. (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 13657 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 13658 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 ) )
 
8.1.8  Natural transformations and the functor category
 
Syntaxcnat 13659 Extend class notation to include the collection of natural transformations.
 class Nat
 
Syntaxcfuc 13660 Extend class notation to include the functor category.
 class FuncCat
 
Definitiondf-nat 13661* 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. (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 13662* Definition of the category of functors between two fixed categories, with the objects being functors and the morphisms being natural transformations. (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 ) ) ) ) )
 >. } )
 
Theoremfnfuc 13663 The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |- FuncCat  Fn  ( Cat  X.  Cat )
 
Theoremnatfval 13664* Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  .x.  =  (comp `  D )   =>    |-  N  =  ( f  e.  ( C  Func  D ) ,  g  e.  ( C  Func  D ) 
 |->  [_ ( 1st `  f
 )  /  r ]_ [_ ( 1st `  g
 )  /  s ]_ { a  e.  X_ x  e.  B  ( ( r `
  x ) J ( s `  x ) )  |  A. x  e.  B  A. y  e.  B  A. h  e.  ( x H y ) ( ( a `
  y ) (
 <. ( r `  x ) ,  ( r `  y ) >.  .x.  (
 s `  y )
 ) ( ( x ( 2nd `  f
 ) y ) `  h ) )  =  ( ( ( x ( 2nd `  g
 ) y ) `  h ) ( <. ( r `  x ) ,  ( s `  x ) >.  .x.  (
 s `  y )
 ) ( a `  x ) ) }
 )
 
Theoremisnat 13665* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  .x.  =  (comp `  D )   &    |-  ( ph  ->  F ( C  Func  D ) G )   &    |-  ( ph  ->  K ( C  Func  D ) L )   =>    |-  ( ph  ->  ( A  e.  ( <. F ,  G >. N <. K ,  L >. )  <->  ( A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) )  /\  A. x  e.  B  A. y  e.  B  A. h  e.  ( x H y ) ( ( A `
  y ) (
 <. ( F `  x ) ,  ( F `  y ) >.  .x.  ( K `  y ) ) ( ( x G y ) `  h ) )  =  (
 ( ( x L y ) `  h ) ( <. ( F `
  x ) ,  ( K `  x ) >.  .x.  ( K `  y ) ) ( A `  x ) ) ) ) )
 
Theoremisnat2 13666* Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  B  =  (
 Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  .x.  =  (comp `  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   =>    |-  ( ph  ->  ( A  e.  ( F N G )  <->  ( A  e.  X_ x  e.  B  ( ( ( 1st `  F ) `  x ) J ( ( 1st `  G ) `  x ) ) 
 /\  A. x  e.  B  A. y  e.  B  A. h  e.  ( x H y ) ( ( A `  y
 ) ( <. ( ( 1st `  F ) `  x ) ,  (
 ( 1st `  F ) `  y ) >.  .x.  (
 ( 1st `  G ) `  y ) ) ( ( x ( 2nd `  F ) y ) `
  h ) )  =  ( ( ( x ( 2nd `  G ) y ) `  h ) ( <. ( ( 1st `  F ) `  x ) ,  ( ( 1st `  G ) `  x ) >.  .x.  ( ( 1st `  G ) `  y ) ) ( A `  x ) ) ) ) )
 
Theoremnatffn 13667 The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   =>    |-  N  Fn  ( ( C  Func  D )  X.  ( C  Func  D ) )
 
Theoremnatrcl 13668 Reverse closure for a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   =>    |-  ( A  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
 
Theoremnat1st2nd 13669 Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  A  e.  ( F N G ) )   =>    |-  ( ph  ->  A  e.  ( <. ( 1st `  F ) ,  ( 2nd `  F ) >. N
 <. ( 1st `  G ) ,  ( 2nd `  G ) >. ) )
 
Theoremnatixp 13670* A natural transformation is a function from the objects of  C to homomorphisms from  F ( x ) to  G ( x ). (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  A  e.  ( <. F ,  G >. N <. K ,  L >. ) )   &    |-  B  =  ( Base `  C )   &    |-  J  =  (  Hom  `  D )   =>    |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) ) )
 
Theoremnatcl 13671 A component of a natural transformation is a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  A  e.  ( <. F ,  G >. N <. K ,  L >. ) )   &    |-  B  =  ( Base `  C )   &    |-  J  =  (  Hom  `  D )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( A `  X )  e.  ( ( F `  X ) J ( K `  X ) ) )
 
Theoremnatfn 13672 A natural transformation is a function on the objects of  C. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  A  e.  ( <. F ,  G >. N <. K ,  L >. ) )   &    |-  B  =  ( Base `  C )   =>    |-  ( ph  ->  A  Fn  B )
 
Theoremnati 13673 Naturality property of a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  N  =  ( C Nat 
 D )   &    |-  ( ph  ->  A  e.  ( <. F ,  G >. N <. K ,  L >. ) )   &    |-  B  =  ( Base `  C )   &    |-  H  =  (  Hom  `  C )   &    |- 
 .x.  =  (comp `  D )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( ( A `  Y ) (
 <. ( F `  X ) ,  ( F `  Y ) >.  .x.  ( K `  Y ) ) ( ( X G Y ) `  R ) )  =  (
 ( ( X L Y ) `  R ) ( <. ( F `
  X ) ,  ( K `  X ) >.  .x.  ( K `  Y ) ) ( A `  X ) ) )
 
Theoremwunnat 13674 A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  C  e.  U )   &    |-  ( ph  ->  D  e.  U )   =>    |-  ( ph  ->  ( C Nat  D )  e.  U )
 
Theoremcatstr 13675 A category structure is a structure. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- 
 { <. ( Base `  ndx ) ,  U >. , 
 <. (  Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .x.  >. } Struct  <. 1 , ; 1
 5 >.
 
Theoremfucval 13676* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  ( C  Func  D )   &    |-  N  =  ( C Nat  D )   &    |-  A  =  ( Base `  C )   &    |-  .x.  =  (comp `  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  ( ph  ->  .xb  =  ( v  e.  ( B  X.  B ) ,  h  e.  B  |->  [_ ( 1st `  v )  /  f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g )  |->  ( x  e.  A  |->  ( ( b `
  x ) (
 <. ( ( 1st `  f
 ) `  x ) ,  ( ( 1st `  g
 ) `  x ) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `
  x ) ) ) ) ) )   =>    |-  ( ph  ->  Q  =  { <. ( Base `  ndx ) ,  B >. , 
 <. (  Hom  `  ndx ) ,  N >. , 
 <. (comp `  ndx ) , 
 .xb  >. } )
 
Theoremfuccofval 13677* Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  ( C  Func  D )   &    |-  N  =  ( C Nat  D )   &    |-  A  =  ( Base `  C )   &    |-  .x.  =  (comp `  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |-  .xb  =  (comp `  Q )   =>    |-  ( ph  ->  .xb  =  ( v  e.  ( B  X.  B ) ,  h  e.  B  |->  [_ ( 1st `  v )  /  f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g )  |->  ( x  e.  A  |->  ( ( b `
  x ) (
 <. ( ( 1st `  f
 ) `  x ) ,  ( ( 1st `  g
 ) `  x ) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `
  x ) ) ) ) ) )
 
Theoremfucbas 13678 The objects of the functor category are functors from  C to  D. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 12-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   =>    |-  ( C  Func  D )  =  ( Base `  Q )
 
Theoremfuchom 13679 The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   =>    |-  N  =  ( 
 Hom  `  Q )
 
Theoremfucco 13680* Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  A  =  ( Base `  C )   &    |-  .x.  =  (comp `  D )   &    |-  .xb  =  (comp `  Q )   &    |-  ( ph  ->  R  e.  ( F N G ) )   &    |-  ( ph  ->  S  e.  ( G N H ) )   =>    |-  ( ph  ->  ( S ( <. F ,  G >.  .xb  H ) R )  =  ( x  e.  A  |->  ( ( S `  x ) ( <. ( ( 1st `  F ) `  x ) ,  ( ( 1st `  G ) `  x ) >.  .x.  (
 ( 1st `  H ) `  x ) ) ( R `  x ) ) ) )
 
Theoremfuccoval 13681 Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  A  =  ( Base `  C )   &    |-  .x.  =  (comp `  D )   &    |-  .xb  =  (comp `  Q )   &    |-  ( ph  ->  R  e.  ( F N G ) )   &    |-  ( ph  ->  S  e.  ( G N H ) )   &    |-  ( ph  ->  X  e.  A )   =>    |-  ( ph  ->  ( ( S ( <. F ,  G >.  .xb  H ) R ) `  X )  =  ( ( S `  X ) (
 <. ( ( 1st `  F ) `  X ) ,  ( ( 1st `  G ) `  X ) >.  .x.  ( ( 1st `  H ) `  X ) ) ( R `  X ) ) )
 
Theoremfuccocl 13682 The composition of two natural transformations is a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  .xb  =  (comp `  Q )   &    |-  ( ph  ->  R  e.  ( F N G ) )   &    |-  ( ph  ->  S  e.  ( G N H ) )   =>    |-  ( ph  ->  ( S ( <. F ,  G >.  .xb  H ) R )  e.  ( F N H ) )
 
Theoremfucidcl 13683 The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  .1.  =  ( Id `  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   =>    |-  ( ph  ->  (  .1.  o.  ( 1st `  F ) )  e.  ( F N F ) )
 
Theoremfuclid 13684 Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  .xb  =  (comp `  Q )   &    |-  .1.  =  ( Id `  D )   &    |-  ( ph  ->  R  e.  ( F N G ) )   =>    |-  ( ph  ->  (
 (  .1.  o.  ( 1st `  G ) ) ( <. F ,  G >. 
 .xb  G ) R )  =  R )
 
Theoremfucrid 13685 Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  .xb  =  (comp `  Q )   &    |-  .1.  =  ( Id `  D )   &    |-  ( ph  ->  R  e.  ( F N G ) )   =>    |-  ( ph  ->  ( R ( <. F ,  F >.  .xb  G ) (  .1.  o.  ( 1st `  F ) ) )  =  R )
 
Theoremfucass 13686 Associativity of natural transformation composition. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  N  =  ( C Nat  D )   &    |-  .xb  =  (comp `  Q )   &    |-  ( ph  ->  R  e.  ( F N G ) )   &    |-  ( ph  ->  S  e.  ( G N H ) )   &    |-  ( ph  ->  T  e.  ( H N K ) )   =>    |-  ( ph  ->  ( ( T ( <. G ,  H >.  .xb  K ) S ) ( <. F ,  G >.  .xb  K ) R )  =  ( T ( <. F ,  H >.  .xb  K ) ( S ( <. F ,  G >.  .xb  H ) R ) ) )
 
Theoremfuccatid 13687* The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   &    |- 
 .1.  =  ( Id `  D )   =>    |-  ( ph  ->  ( Q  e.  Cat  /\  ( Id `  Q )  =  ( f  e.  ( C  Func  D )  |->  (  .1.  o.  ( 1st `  f ) ) ) ) )
 
Theoremfuccat 13688 The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  Q  e.  Cat )
 
Theoremfucid 13689 The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  I  =  ( Id `  Q )   &    |-  .1.  =  ( Id `  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   =>    |-  ( ph  ->  ( I `  F )  =  (  .1.  o.  ( 1st `  F ) ) )
 
Theoremfucsect 13690* Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  (
 Base `  C )   &    |-  N  =  ( C Nat  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  S  =  (Sect `  Q )   &    |-  T  =  (Sect `  D )   =>    |-  ( ph  ->  ( U ( F S G ) V  <->  ( U  e.  ( F N G ) 
 /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x ) T ( ( 1st `  G ) `  x ) ) ( V `
  x ) ) ) )
 
Theoremfucinv 13691* Two natural transformations are inverses of each other iff all the components are inverse. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  (
 Base `  C )   &    |-  N  =  ( C Nat  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  I  =  (Inv `  Q )   &    |-  J  =  (Inv `  D )   =>    |-  ( ph  ->  ( U ( F I G ) V  <->  ( U  e.  ( F N G ) 
 /\  V  e.  ( G N F )  /\  A. x  e.  B  ( U `  x ) ( ( ( 1st `  F ) `  x ) J ( ( 1st `  G ) `  x ) ) ( V `
  x ) ) ) )
 
Theoreminvfuc 13692* If  V (
x ) is an inverse to  U ( x ) for each  x, and  U is a natural transformation, then  V is also a natural transformation and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  (
 Base `  C )   &    |-  N  =  ( C Nat  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  I  =  (Inv `  Q )   &    |-  J  =  (Inv `  D )   &    |-  ( ph  ->  U  e.  ( F N G ) )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( U `  x ) ( ( ( 1st `  F ) `  x ) J ( ( 1st `  G ) `  x ) ) X )   =>    |-  ( ph  ->  U ( F I G ) ( x  e.  B  |->  X ) )
 
Theoremfuciso 13693* A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Q  =  ( C FuncCat  D )   &    |-  B  =  (
 Base `  C )   &    |-  N  =  ( C Nat  D )   &    |-  ( ph  ->  F  e.  ( C  Func  D ) )   &    |-  ( ph  ->  G  e.  ( C  Func  D ) )   &    |-  I  =  ( 
 Iso  `  Q )   &    |-  J  =  (  Iso  `  D )   =>    |-  ( ph  ->  ( A  e.  ( F I G )  <->  ( A  e.  ( F N G ) 
 /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x ) J ( ( 1st `  G ) `  x ) ) ) ) )
 
Theoremnatpropd 13694 If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  A )  =  ( 
 Homf  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  Cat )   &    |-  ( ph  ->  B  e.  Cat )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  ( A Nat  C )  =  ( B Nat  D ) )
 
Theoremfucpropd 13695 If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( ph  ->  (  Homf  `  A )  =  ( 
 Homf  `  B ) )   &    |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )   &    |-  ( ph  ->  (  Homf  `  C )  =  (  Homf  `  D ) )   &    |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )   &    |-  ( ph  ->  A  e.  Cat )   &    |-  ( ph  ->  B  e.  Cat )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  D  e.  Cat )   =>    |-  ( ph  ->  ( A FuncCat  C )  =  ( B FuncCat  D ) )
 
8.2  Arrows (disjointified hom-sets)
 
Syntaxcdoma 13696 Extend class notation to include the domain extractor for an arrow.
 class domA
 
Syntaxccoda 13697 Extend class notation to include the codomain extractor for an arrow.
 class coda
 
Syntaxcarw 13698 Extend class notation to include the collection of all arrows of a category.
 class Nat
 
Syntaxchoma 13699 Extend class notation to include the set of all arrows with a specific domain and codomain.
 class Homa
 
Definitiondf-doma 13700 Definition of the domain extractor for an arrow. (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
 |- domA  =  ( 1st  o.  1st )
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