defpred S₁[ object , object ] means for f being Morphism of C st f = $1 holds
$2 = [[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))];
set X = the carrier' of C;
set Y = Maps (Hom C);
A6: for x being object st x in the carrier' of C holds
ex y being object st
( y in Maps (Hom C) & S₁[x,y] )

proof

let x be object ; :: thesis: ( x in the carrier' of C implies ex y being object st
( y in Maps (Hom C) & S₁[x,y] ) )
assume x in the carrier' of C ; :: thesis: ex y being object st
( y in Maps (Hom C) & S₁[x,y] )
then reconsider f = x as Morphism of C ;
take y = [[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))]; :: thesis: ( y in Maps (Hom C) & S₁[x,y] )
y is Element of Maps (Hom C) by Th47;
hence ( y in Maps (Hom C) & S₁[x,y] ) ; :: thesis: verum

end;

consider h being Function such that
A7: ( dom h = the carrier' of C & rng h c= Maps (Hom C) ) and
A8: for x being object st x in the carrier' of C holds
S₁[x,h . x] from FUNCT_1:sch 6(A6);
reconsider h = h as Function of the carrier' of C,(Maps (Hom C)) by A7, FUNCT_2:def 1, RELSET_1:4;
take h ; :: thesis: for f being Morphism of C holds h . f = [[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))]
thus for f being Morphism of C holds h . f = [[(Hom ((cod f),a)),(Hom ((dom f),a))],(hom (f,a))] by A8; :: thesis: verum