let C, D be Category; :: thesis: for S being Contravariant_Functor of C,D holds S *' is Functor of C,D opp
let S be Contravariant_Functor of C,D; :: thesis: S *' is Functor of C,D opp

now

thus for c being Object of C ex d being Object of (D opp) st (S *') . (id c) = id d :: thesis: ( ( for f being Morphism of C holds
( (S *') . (id (dom f)) = id (dom ((S *') . f)) & (S *') . (id (cod f)) = id (cod ((S *') . f)) ) ) & ( for f, g being Morphism of C st dom g = cod f holds
(S *') . (g * f) = ((S *') . g) * ((S *') . f) ) )

proof

let c be Object of C; :: thesis: ex d being Object of (D opp) st (S *') . (id c) = id d
(S *') . (id c) = id (((Obj S) . c) opp) by Lm13;
hence ex d being Object of (D opp) st (S *') . (id c) = id d ; :: thesis: verum

end;

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

proof

let f be Morphism of C; :: thesis: ( (S *') . (id (dom f)) = id (dom ((S *') . f)) & (S *') . (id (cod f)) = id (cod ((S *') . f)) )
thus (S *') . (id (dom f)) = id ((Obj (S *')) . (dom f)) by Lm18
.= id (dom ((S *') . f)) by Lm19 ; :: thesis: (S *') . (id (cod f)) = id (cod ((S *') . f))
thus (S *') . (id (cod f)) = id ((Obj (S *')) . (cod f)) by Lm18
.= id (cod ((S *') . f)) by Lm19 ; :: thesis: verum

end;

let f, g be Morphism of C; :: thesis: ( dom g = cod f implies (S *') . (g * f) = ((S *') . g) * ((S *') . f) )
assume A1: dom g = cod f ; :: thesis: (S *') . (g * f) = ((S *') . g) * ((S *') . f)
then A2: dom (S . f) = cod (S . g) by Th34;
thus (S *') . (g * f) = (S . (g * f)) opp by Def9
.= ((S . f) * (S . g)) opp by A1, Def7
.= ((S . g) opp) * ((S . f) opp) by A2, Th17
.= ((S *') . g) * ((S . f) opp) by Def9
.= ((S *') . g) * ((S *') . f) by Def9 ; :: thesis: verum

end;

hence S *' is Functor of C,D opp by CAT_1:61; :: thesis: verum