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Theorem List for Metamath Proof Explorer - 13701-13800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisfunc 13701* 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 13702* 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 13703 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 13704* 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 13705 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 13706 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 13707 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 13708 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 13709 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 13710 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 13711 The image of an isomorphism under a functor is an isomorphism. (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 13712 A functor on categories yields a functor on the opposite categories (in the same direction). (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 13713* Value of the composition of two functors. (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 13714 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 13715 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 13716 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 13717 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 13718 The identity functor is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  I  =  (idfunc `  C )   =>    |-  ( C  e.  Cat  ->  I  e.  ( C  Func  C ) )
 
Theoremcofuval 13719* 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 13720 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 13721 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 13722 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 13723 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 13724* 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 13725 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 13726 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 13727 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 13728 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 13729* 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 13730* 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 13731 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 13732 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 13733 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 13734* 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 13735 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 13736 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 13737 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 13738 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 13739 Extend class notation with the class of all full functors.
 class Full
 
Syntaxcfth 13740 Extend class notation with the class of all faithful functors.
 class Faith
 
Definitiondf-full 13741* 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 13742* 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 13743 A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( C Full  D ) 
 C_  ( C  Func  D )
 
Theoremfthfunc 13744 A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |-  ( C Faith  D ) 
 C_  ( C  Func  D )
 
Theoremrelfull 13745 The set of full functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |- 
 Rel  ( C Full  D )
 
Theoremrelfth 13746 The set of faithful functors is a relation. (Contributed by Mario Carneiro, 26-Jan-2017.)
 |- 
 Rel  ( C Faith  D )
 
Theoremisfull 13747* 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 13748* 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 13749 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 13750* 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 13751* 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 13752* 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 13753* 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 13754 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 13755 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 13756 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 13757 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 13758 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 13759 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 13760 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 13761 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 13762 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 13763 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 13764 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 13765 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 13766 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 13767* 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 13768 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 13769 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 13770 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 13771 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 13772 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 13773 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 13774 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 13775 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 13776 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 13777 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 13778 Extend class notation to include the collection of natural transformations.
 class Nat
 
Syntaxcfuc 13779 Extend class notation to include the functor category.
 class FuncCat
 
Definitiondf-nat 13780* 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 13781* 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 13782 The FuncCat operation is a well-defined function on categories. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |- FuncCat  Fn  ( Cat  X.  Cat )
 
Theoremnatfval 13783* 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 13784* 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 13785* 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 13786 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 13787 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 13788 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 13789* 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 13790 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 13791 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 13792 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 13793 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 13794 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 13795* 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 13796* 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 13797 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 13798 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 13799* 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 13800 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 ) ) )
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