reserve V for non empty set,
  A,B,A9,B9 for Element of V;
reserve f,f9 for Element of Funcs(V);
reserve m,m1,m2,m3,m9 for Element of Maps V;
reserve a,b for Object of Ens(V);
reserve f,g,f1,f2 for Morphism of Ens(V);

theorem
  for W being non empty Subset of V
   holds Ens W is full Subcategory of Ens V
proof
  let W be non empty Subset of V;
   reconsider E = Ens W as Subcategory of Ens V by Lm7;
 for a,b being Object of E, a9,b9 being Object of Ens(V) st a = a9 &
  b = b9 holds Hom(a,b) = Hom(a9,b9)
  proof
    let a,b be Object of E, a9,b9 be Object of Ens(V);
    assume
A1: a = a9 & b = b9;
    reconsider aa=a, bb=b as Element of Ens W;
    Hom(aa,bb) = Maps(@aa,@bb) & Hom(a9,b9) = Maps(@a9,@b9) by Th26;
    hence thesis by A1,Lm5;
  end;
  hence thesis by CAT_2:def 6;
end;
