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 Th82:
  for T being Functor of C,D for c,c9 being Object of C holds T.:
  Hom(c,c9) c= Hom(T.c,T.c9)
proof
  let T be Functor of C,D;
  let c,c9 be Object of C;
  let f be object;
  assume f in T.:Hom(c,c9);
  then ex g being Element of the carrier' of C st g in Hom(c,c9) & f = T.g by
FUNCT_2:65;
  hence thesis by Th76;
end;
