let C be Category; :: thesis: for a, c being Object of C holds hom ((id c),a) = id (Hom (c,a))

let a, c be Object of C; :: thesis: hom ((id c),a) = id (Hom (c,a))

set A = Hom (c,a);

let a, c be Object of C; :: thesis: hom ((id c),a) = id (Hom (c,a))

set A = Hom (c,a);

now :: thesis: ( dom (hom ((id c),a)) = Hom (c,a) & ( for x being object st x in Hom (c,a) holds

(hom ((id c),a)) . x = x ) )

hence
hom ((id c),a) = id (Hom (c,a))
by FUNCT_1:17; :: thesis: verum(hom ((id c),a)) . x = x ) )

( Hom (c,a) = {} implies Hom (c,a) = {} )
;

hence dom (hom ((id c),a)) = Hom (c,a) by FUNCT_2:def 1; :: thesis: for x being object st x in Hom (c,a) holds

(hom ((id c),a)) . x = x

let x be object ; :: thesis: ( x in Hom (c,a) implies (hom ((id c),a)) . x = x )

assume A1: x in Hom (c,a) ; :: thesis: (hom ((id c),a)) . x = x

then reconsider g = x as Morphism of C ;

A2: dom g = c by A1, CAT_1:1;

thus (hom ((id c),a)) . x = g (*) (id c) by A1, Def19

.= x by A2, CAT_1:22 ; :: thesis: verum

end;hence dom (hom ((id c),a)) = Hom (c,a) by FUNCT_2:def 1; :: thesis: for x being object st x in Hom (c,a) holds

(hom ((id c),a)) . x = x

let x be object ; :: thesis: ( x in Hom (c,a) implies (hom ((id c),a)) . x = x )

assume A1: x in Hom (c,a) ; :: thesis: (hom ((id c),a)) . x = x

then reconsider g = x as Morphism of C ;

A2: dom g = c by A1, CAT_1:1;

thus (hom ((id c),a)) . x = g (*) (id c) by A1, Def19

.= x by A2, CAT_1:22 ; :: thesis: verum