reserve C for Category,
  C1,C2 for Subcategory of C;

theorem
  for C being Category, a,b being Object of C holds
  Hom(a,b) = (a Hom) /\ (Hom b)
proof
  let C be Category, a,b be Object of C;
  hereby
    let x be object;
    assume
A1: x in Hom(a,b);
    then reconsider f = x as Morphism of C;
A2: dom f = a by A1,CAT_1:1;
A3: cod f = b by A1,CAT_1:1;
A4: f in a Hom by A2,Th24;
    f in Hom b by A3,Th23;
    hence x in (a Hom) /\ (Hom b) by A4,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A5: x in (a Hom) /\ (Hom b);
  then
A6: x in a Hom by XBOOLE_0:def 4;
A7: x in Hom b by A5,XBOOLE_0:def 4;
  reconsider f = x as Morphism of C by A5;
A8: dom f = a by A6,Th24;
  cod f = b by A7,Th23;
  hence thesis by A8;
end;
