defpred S₁[ object , object ] means for g being Morphism of C st g = $1 holds
$2 = f (*) g;
set X = Hom (a,(dom f));
set Y = Hom (a,(cod f));
A1: for x being object st x in Hom (a,(dom f)) holds
ex y being object st
( y in Hom (a,(cod f)) & S₁[x,y] )

proof

let x be object ; :: thesis: ( x in Hom (a,(dom f)) implies ex y being object st
( y in Hom (a,(cod f)) & S₁[x,y] ) )
assume A2: x in Hom (a,(dom f)) ; :: thesis: ex y being object st
( y in Hom (a,(cod f)) & S₁[x,y] )
then reconsider g = x as Morphism of a, dom f by CAT_1:def 5;
take f (*) g ; :: thesis: ( f (*) g in Hom (a,(cod f)) & S₁[x,f (*) g] )
( Hom ((dom f),(cod f)) <> {} & f is Morphism of dom f, cod f ) by CAT_1:1, CAT_1:4;
hence ( f (*) g in Hom (a,(cod f)) & S₁[x,f (*) g] ) by A2, CAT_1:23; :: thesis: verum

end;

consider h being Function such that
A3: ( dom h = Hom (a,(dom f)) & rng h c= Hom (a,(cod f)) ) and
A4: for x being object st x in Hom (a,(dom f)) holds
S₁[x,h . x] from FUNCT_1:sch 6(A1);
Hom ((dom f),(cod f)) <> {} by CAT_1:2;
then ( Hom (a,(cod f)) = {} implies Hom (a,(dom f)) = {} ) by CAT_1:24;
then reconsider h = h as Function of (Hom (a,(dom f))),(Hom (a,(cod f))) by A3, FUNCT_2:def 1, RELSET_1:4;
take h ; :: thesis: for g being Morphism of C st g in Hom (a,(dom f)) holds
h . g = f (*) g
thus for g being Morphism of C st g in Hom (a,(dom f)) holds
h . g = f (*) g by A4; :: thesis: verum