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 Contravariant_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 Contravariant_Functor of C, Ens V
let a be Object of C; :: thesis: ( Hom C c= V implies hom-? a is Contravariant_Functor of C, Ens V )
assume A1: Hom C c= V ; :: thesis: hom-? a is Contravariant_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 (cod (T . f)) & T . (id (cod f)) = id (dom (T . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds
T . (g (*) f) = (T . f) (*) (T . g) ) )

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

proof

let c be Object of C; :: thesis: ex d being Object of (Ens V) st T . (id c) = id d
Hom (c,a) in Hom C ;
then reconsider A = Hom (c,a) as Element of V by A1;
take d = @ A; :: thesis: T . (id c) = id d
thus T . (id c) = id d by A1, Lm10; :: thesis: verum

end;

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

proof

let f be Morphism of C; :: thesis: ( T . (id (dom f)) = id (cod (T . f)) & T . (id (cod f)) = id (dom (T . f)) )
set b = cod f;
set c = dom f;
set g = T . f;
( Hom ((cod f),a) in Hom C & Hom ((dom f),a) in Hom C ) ;
then reconsider A = Hom ((cod f),a), B = Hom ((dom f),a) as Element of V by A1;
A2: [[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))] = @ (T . f) by Def21
.= [[(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 (@ B) by A1, Lm10
.= id (cod (T . f)) by A2, Lm1 ; :: thesis: T . (id (cod f)) = id (dom (T . f))
thus T . (id (cod f)) = id (@ A) by A1, Lm10
.= id (dom (T . f)) by A2, Lm1 ; :: thesis: verum

end;

let f, g be Morphism of C; :: thesis: ( dom g = cod f implies T . (g (*) f) = (T . f) (*) (T . g) )
assume A3: dom g = cod f ; :: thesis: T . (g (*) f) = (T . f) (*) (T . g)
A4: [[(Hom ((cod g),a)),(Hom ((dom g),a))],(hom (g,a))] = @ (T . g) by Def21
.= [[(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 (g,a) by XTUPLE_0:1;
dom (T . g) = Hom ((cod g),a) by A4, Lm1;
then A6: dom (@ (T . g)) = Hom ((cod g),a) by Def9;
A7: cod (T . g) = Hom ((dom g),a) by A4, Lm1;
then A8: cod (@ (T . g)) = Hom ((dom g),a) by Def10;
A9: [[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))] = @ (T . f) by Def21
.= [[(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 (f,a) by XTUPLE_0:1;
cod (T . f) = Hom ((dom f),a) by A9, Lm1;
then A11: cod (@ (T . f)) = Hom ((dom f),a) by Def10;
A12: dom (T . f) = Hom ((cod f),a) by A9, Lm1;
then A13: dom (@ (T . f)) = Hom ((cod f),a) by Def9;
( dom (g (*) f) = dom f & cod (g (*) f) = cod g ) by A3, CAT_1:17;
hence T . (g (*) f) = [[(Hom ((cod g),a)),(Hom ((dom f),a))],(hom ((g (*) f),a))] by Def21
.= [[(Hom ((cod g),a)),(Hom ((dom f),a))],((hom (f,a)) * (hom (g,a)))] by A3, Th45
.= (@ (T . f)) * (@ (T . g)) by A3, A10, A13, A11, A5, A6, A8, Def6
.= (T . f) (*) (T . g) by A3, A12, A7, Th27 ;
:: thesis: verum

end;

hence hom-? a is Contravariant_Functor of C, Ens V by OPPCAT_1:def 9; :: thesis: verum