reserve C for CatStr;
reserve f,g for Morphism of C;
reserve C for non void non empty CatStr,
  f,g for Morphism of C,
  a,b,c,d for Object of C;
reserve o,m for set;
reserve B,C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,f1,f2,g,g1,g2 for Morphism of C;
reserve f,f1,f2 for Morphism of a,b;
reserve f9 for Morphism of b,a;
reserve g for Morphism of b,c;
reserve h,h1,h2 for Morphism of c,d;

theorem
  for T being Functor of C,D holds T is full iff for c,c9 being Object
  of C holds rng hom(T,c,c9) = Hom(T.c,T.c9)
proof
  let T be Functor of C,D;
  thus T is full implies for c,c9 being Object of C holds rng hom(T,c,c9) =
  Hom(T.c,T.c9)
  proof
    assume
A1: T is full;
    let c,c9 be Object of C;
    now
      assume
A2:   Hom(T.c,T.c9) <> {};
      for g being object holds g in rng hom(T,c,c9) iff g in Hom(T.c,T.c9)
      proof
       let g be object;
       thus g in rng hom(T,c,c9) implies g in Hom(T.c,T.c9)
        proof
         assume g in rng hom(T,c,c9);
         then consider f being object such that
A3:       f in dom hom(T,c,c9) and
A4:       hom(T,c,c9).f = g by FUNCT_1:def 3;
          f in Hom(c,c9) by A2,A3,FUNCT_2:def 1;
         hence thesis by A2,A4,FUNCT_2:5;
        end;
        assume g in Hom(T.c,T.c9);
        then g is Morphism of T.c,T.c9 by Def3;
        then consider f being Morphism of c,c9 such that
A5:     g = T.f by A1,A2;
        Hom(c,c9) <> {} by A1,A2;
        then f in Hom(c,c9) by Def3;
        then hom(T,c,c9).f in rng hom(T,c,c9) by A2,FUNCT_2:4;
       hence thesis by A5,A1,A2,Th83;
      end;
      hence thesis by TARSKI:2;
    end;
    hence thesis;
  end;
  assume
A6: for c,c9 being Object of C holds rng hom(T,c,c9) = Hom(T.c,T.c9);
  let c,c9 be Object of C such that
A7: Hom(T.c,T.c9) <> {};
  let g be Morphism of T.c,T.c9;
  g in Hom(T.c,T.c9) by A7,Def3;
  then g in rng hom(T,c,c9) by A6;
  then consider f being object such that
A8: f in dom hom(T,c,c9) and
A9: hom(T,c,c9).f = g by FUNCT_1:def 3;
  thus Hom(c,c9) <> {} by A8;
A10: f in Hom(c,c9) by A7,A8,FUNCT_2:def 1;
  then reconsider f as Morphism of c,c9 by Def3;
  take f;
  thus thesis by A9,A10,Th83;
end;
