let B, C, D be Category; :: thesis: for S1 being Contravariant_Functor of C,B
for S2 being Contravariant_Functor of B,D holds S2 * S1 is Functor of C,D
let S1 be Contravariant_Functor of C,B; :: thesis: for S2 being Contravariant_Functor of B,D holds S2 * S1 is Functor of C,D
let S2 be Contravariant_Functor of B,D; :: thesis: S2 * S1 is Functor of C,D
set T = S2 * S1;
now :: thesis: ( ( for c being Object of C ex d being Object of D st (S2 * S1) . (id c) = id d ) & ( for f being Morphism of C holds
( (S2 * S1) . (id (dom f)) = id (dom ((S2 * S1) . f)) & (S2 * S1) . (id (cod f)) = id (cod ((S2 * S1) . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds
(S2 * S1) . (g (*) f) = ((S2 * S1) . g) (*) ((S2 * S1) . f) ) )
thus for c being Object of C ex d being Object of D st (S2 * S1) . (id c) = id d :: thesis: ( ( for f being Morphism of C holds
( (S2 * S1) . (id (dom f)) = id (dom ((S2 * S1) . f)) & (S2 * S1) . (id (cod f)) = id (cod ((S2 * S1) . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds
(S2 * S1) . (g (*) f) = ((S2 * S1) . g) (*) ((S2 * S1) . f) ) )
proof
let c be Object of C; :: thesis: ex d being Object of D st (S2 * S1) . (id c) = id d
consider b being Object of B such that
A1: S1 . (id c) = id b by Def9;
consider d being Object of D such that
A2: S2 . (id b) = id d by Def9;
take d ; :: thesis: (S2 * S1) . (id c) = id d
thus (S2 * S1) . (id c) = id d by A1, A2, FUNCT_2:15; :: thesis: verum
end;
thus for f being Morphism of C holds
( (S2 * S1) . (id (dom f)) = id (dom ((S2 * S1) . f)) & (S2 * S1) . (id (cod f)) = id (cod ((S2 * S1) . f)) ) :: thesis: for f, g being Morphism of C st dom g = cod f holds
(S2 * S1) . (g (*) f) = ((S2 * S1) . g) (*) ((S2 * S1) . f)
proof
let f be Morphism of C; :: thesis: ( (S2 * S1) . (id (dom f)) = id (dom ((S2 * S1) . f)) & (S2 * S1) . (id (cod f)) = id (cod ((S2 * S1) . f)) )
thus (S2 * S1) . (id (dom f)) = S2 . (S1 . (id (dom f))) by FUNCT_2:15
.= S2 . (id (cod (S1 . f))) by Def9
.= id (dom (S2 . (S1 . f))) by Def9
.= id (dom ((S2 * S1) . f)) by FUNCT_2:15 ; :: thesis: (S2 * S1) . (id (cod f)) = id (cod ((S2 * S1) . f))
thus (S2 * S1) . (id (cod f)) = S2 . (S1 . (id (cod f))) by FUNCT_2:15
.= S2 . (id (dom (S1 . f))) by Def9
.= id (cod (S2 . (S1 . f))) by Def9
.= id (cod ((S2 * S1) . f)) by FUNCT_2:15 ; :: thesis: verum
end;
let f, g be Morphism of C; :: thesis: ( dom g = cod f implies (S2 * S1) . (g (*) f) = ((S2 * S1) . g) (*) ((S2 * S1) . f) )
assume A3: dom g = cod f ; :: thesis: (S2 * S1) . (g (*) f) = ((S2 * S1) . g) (*) ((S2 * S1) . f)
then A4: cod (S1 . g) = dom (S1 . f) by Th31;
thus (S2 * S1) . (g (*) f) = S2 . (S1 . (g (*) f)) by FUNCT_2:15
.= S2 . ((S1 . f) (*) (S1 . g)) by A3, Def9
.= (S2 . (S1 . g)) (*) (S2 . (S1 . f)) by A4, Def9
.= ((S2 * S1) . g) (*) (S2 . (S1 . f)) by FUNCT_2:15
.= ((S2 * S1) . g) (*) ((S2 * S1) . f) by FUNCT_2:15 ; :: thesis: verum
end;
hence S2 * S1 is Functor of C,D by CAT_1:61; :: thesis: verum