let V be non empty set ; :: thesis: for C being Category

for a being Object of C st Hom C c= V holds

hom?- a is Functor of C, Ens V

let C be Category; :: thesis: for a being Object of C st Hom C c= V holds

hom?- a is Functor of C, Ens V

let a be Object of C; :: thesis: ( Hom C c= V implies hom?- a is Functor of C, Ens V )

assume A1: Hom C c= V ; :: thesis: hom?- a is Functor of C, Ens V

then reconsider T = hom?- a as Function of the carrier' of C, the carrier' of (Ens V) by Lm8;

for a being Object of C st Hom C c= V holds

hom?- a is Functor of C, Ens V

let C be Category; :: thesis: for a being Object of C st Hom C c= V holds

hom?- a is Functor of C, Ens V

let a be Object of C; :: thesis: ( Hom C c= V implies hom?- a is Functor of C, Ens V )

assume A1: Hom C c= V ; :: thesis: hom?- a is Functor of C, Ens V

then reconsider T = hom?- a as Function of the carrier' of C, the carrier' of (Ens V) by Lm8;

now :: thesis: ( ( for c being Object of C ex d being Object of (Ens V) st T . (id c) = id d ) & ( for f being Morphism of C holds

( T . (id (dom f)) = id (dom (T . f)) & T . (id (cod f)) = id (cod (T . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds

T . (g (*) f) = (T . g) (*) (T . f) ) )

hence
hom?- a is Functor of C, Ens V
by CAT_1:61; :: thesis: verum( T . (id (dom f)) = id (dom (T . f)) & T . (id (cod f)) = id (cod (T . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds

T . (g (*) f) = (T . g) (*) (T . f) ) )

thus
for c being Object of C ex d being Object of (Ens V) st T . (id c) = id d
:: thesis: ( ( for f being Morphism of C holds

( T . (id (dom f)) = id (dom (T . f)) & T . (id (cod f)) = id (cod (T . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds

T . (g (*) f) = (T . g) (*) (T . f) ) )

( T . (id (dom f)) = id (dom (T . f)) & T . (id (cod f)) = id (cod (T . f)) ) :: thesis: for f, g being Morphism of C st dom g = cod f holds

T . (g (*) f) = (T . g) (*) (T . f)

assume A3: dom g = cod f ; :: thesis: T . (g (*) f) = (T . g) (*) (T . f)

A4: [[(Hom (a,(dom g))),(Hom (a,(cod g)))],(hom (a,g))] = @ (T . g) by Def20

.= [[(dom (@ (T . g))),(cod (@ (T . g)))],((@ (T . g)) `2)] by Th8

.= [[(dom (T . g)),(cod (@ (T . g)))],((@ (T . g)) `2)] by Def9

.= [[(dom (T . g)),(cod (T . g))],((@ (T . g)) `2)] by Def10 ;

then A5: (@ (T . g)) `2 = hom (a,g) by XTUPLE_0:1;

cod (T . g) = Hom (a,(cod g)) by A4, Lm1;

then A6: cod (@ (T . g)) = Hom (a,(cod g)) by Def10;

A7: dom (T . g) = Hom (a,(dom g)) by A4, Lm1;

then A8: dom (@ (T . g)) = Hom (a,(dom g)) by Def9;

A9: [[(Hom (a,(dom f))),(Hom (a,(cod f)))],(hom (a,f))] = @ (T . f) by Def20

.= [[(dom (@ (T . f))),(cod (@ (T . f)))],((@ (T . f)) `2)] by Th8

.= [[(dom (T . f)),(cod (@ (T . f)))],((@ (T . f)) `2)] by Def9

.= [[(dom (T . f)),(cod (T . f))],((@ (T . f)) `2)] by Def10 ;

then A10: (@ (T . f)) `2 = hom (a,f) by XTUPLE_0:1;

dom (T . f) = Hom (a,(dom f)) by A9, Lm1;

then A11: dom (@ (T . f)) = Hom (a,(dom f)) by Def9;

A12: cod (T . f) = Hom (a,(cod f)) by A9, Lm1;

then A13: cod (@ (T . f)) = Hom (a,(cod f)) by Def10;

( dom (g (*) f) = dom f & cod (g (*) f) = cod g ) by A3, CAT_1:17;

hence T . (g (*) f) = [[(Hom (a,(dom f))),(Hom (a,(cod g)))],(hom (a,(g (*) f)))] by Def20

.= [[(Hom (a,(dom f))),(Hom (a,(cod g)))],((hom (a,g)) * (hom (a,f)))] by A3, Th44

.= (@ (T . g)) * (@ (T . f)) by A3, A10, A11, A13, A5, A8, A6, Def6

.= (T . g) (*) (T . f) by A3, A12, A7, Th27 ;

:: thesis: verum

end;( T . (id (dom f)) = id (dom (T . f)) & T . (id (cod f)) = id (cod (T . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds

T . (g (*) f) = (T . g) (*) (T . f) ) )

proof

thus
for f being Morphism of C holds
let c be Object of C; :: thesis: ex d being Object of (Ens V) st T . (id c) = id d

Hom (a,c) in Hom C ;

then reconsider A = Hom (a,c) as Element of V by A1;

take d = @ A; :: thesis: T . (id c) = id d

thus T . (id c) = id d by A1, Lm9; :: thesis: verum

end;Hom (a,c) in Hom C ;

then reconsider A = Hom (a,c) as Element of V by A1;

take d = @ A; :: thesis: T . (id c) = id d

thus T . (id c) = id d by A1, Lm9; :: thesis: verum

( T . (id (dom f)) = id (dom (T . f)) & T . (id (cod f)) = id (cod (T . f)) ) :: thesis: for f, g being Morphism of C st dom g = cod f holds

T . (g (*) f) = (T . g) (*) (T . f)

proof

let f, g be Morphism of C; :: thesis: ( dom g = cod f implies T . (g (*) f) = (T . g) (*) (T . f) )
let f be Morphism of C; :: thesis: ( T . (id (dom f)) = id (dom (T . f)) & T . (id (cod f)) = id (cod (T . f)) )

set b = dom f;

set c = cod f;

set g = T . f;

( Hom (a,(dom f)) in Hom C & Hom (a,(cod f)) in Hom C ) ;

then reconsider A = Hom (a,(dom f)), B = Hom (a,(cod f)) as Element of V by A1;

A2: [[(Hom (a,(dom f))),(Hom (a,(cod f)))],(hom (a,f))] = @ (T . f) by Def20

.= [[(dom (@ (T . f))),(cod (@ (T . f)))],((@ (T . f)) `2)] by Th8

.= [[(dom (T . f)),(cod (@ (T . f)))],((@ (T . f)) `2)] by Def9

.= [[(dom (T . f)),(cod (T . f))],((@ (T . f)) `2)] by Def10 ;

thus T . (id (dom f)) = id (@ A) by A1, Lm9

.= id (dom (T . f)) by A2, Lm1 ; :: thesis: T . (id (cod f)) = id (cod (T . f))

thus T . (id (cod f)) = id (@ B) by A1, Lm9

.= id (cod (T . f)) by A2, Lm1 ; :: thesis: verum

end;set b = dom f;

set c = cod f;

set g = T . f;

( Hom (a,(dom f)) in Hom C & Hom (a,(cod f)) in Hom C ) ;

then reconsider A = Hom (a,(dom f)), B = Hom (a,(cod f)) as Element of V by A1;

A2: [[(Hom (a,(dom f))),(Hom (a,(cod f)))],(hom (a,f))] = @ (T . f) by Def20

.= [[(dom (@ (T . f))),(cod (@ (T . f)))],((@ (T . f)) `2)] by Th8

.= [[(dom (T . f)),(cod (@ (T . f)))],((@ (T . f)) `2)] by Def9

.= [[(dom (T . f)),(cod (T . f))],((@ (T . f)) `2)] by Def10 ;

thus T . (id (dom f)) = id (@ A) by A1, Lm9

.= id (dom (T . f)) by A2, Lm1 ; :: thesis: T . (id (cod f)) = id (cod (T . f))

thus T . (id (cod f)) = id (@ B) by A1, Lm9

.= id (cod (T . f)) by A2, Lm1 ; :: thesis: verum

assume A3: dom g = cod f ; :: thesis: T . (g (*) f) = (T . g) (*) (T . f)

A4: [[(Hom (a,(dom g))),(Hom (a,(cod g)))],(hom (a,g))] = @ (T . g) by Def20

.= [[(dom (@ (T . g))),(cod (@ (T . g)))],((@ (T . g)) `2)] by Th8

.= [[(dom (T . g)),(cod (@ (T . g)))],((@ (T . g)) `2)] by Def9

.= [[(dom (T . g)),(cod (T . g))],((@ (T . g)) `2)] by Def10 ;

then A5: (@ (T . g)) `2 = hom (a,g) by XTUPLE_0:1;

cod (T . g) = Hom (a,(cod g)) by A4, Lm1;

then A6: cod (@ (T . g)) = Hom (a,(cod g)) by Def10;

A7: dom (T . g) = Hom (a,(dom g)) by A4, Lm1;

then A8: dom (@ (T . g)) = Hom (a,(dom g)) by Def9;

A9: [[(Hom (a,(dom f))),(Hom (a,(cod f)))],(hom (a,f))] = @ (T . f) by Def20

.= [[(dom (@ (T . f))),(cod (@ (T . f)))],((@ (T . f)) `2)] by Th8

.= [[(dom (T . f)),(cod (@ (T . f)))],((@ (T . f)) `2)] by Def9

.= [[(dom (T . f)),(cod (T . f))],((@ (T . f)) `2)] by Def10 ;

then A10: (@ (T . f)) `2 = hom (a,f) by XTUPLE_0:1;

dom (T . f) = Hom (a,(dom f)) by A9, Lm1;

then A11: dom (@ (T . f)) = Hom (a,(dom f)) by Def9;

A12: cod (T . f) = Hom (a,(cod f)) by A9, Lm1;

then A13: cod (@ (T . f)) = Hom (a,(cod f)) by Def10;

( dom (g (*) f) = dom f & cod (g (*) f) = cod g ) by A3, CAT_1:17;

hence T . (g (*) f) = [[(Hom (a,(dom f))),(Hom (a,(cod g)))],(hom (a,(g (*) f)))] by Def20

.= [[(Hom (a,(dom f))),(Hom (a,(cod g)))],((hom (a,g)) * (hom (a,f)))] by A3, Th44

.= (@ (T . g)) * (@ (T . f)) by A3, A10, A11, A13, A5, A8, A6, Def6

.= (T . g) (*) (T . f) by A3, A12, A7, Th27 ;

:: thesis: verum