reserve I for set,
  x,x1,x2,y,z for set,
  A for non empty set;
reserve C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,g,h,i,j,k,p1,p2,q1,q2,i1,i2,j1,j2 for Morphism of C;
reserve f for Morphism of a,b,
        g for Morphism of b,a;
reserve g for Morphism of b,c;

theorem
  for T being Functor of C,D st f is coretraction holds T/.f is coretraction
proof
  let T be Functor of C,D;
  assume
A1: Hom(a,b) <> {} & Hom(b,a) <> {};
  given i being Morphism of b,a such that
A2: i*f = id a;
  thus Hom(T.a,T.b) <> {} & Hom(T.b,T.a) <> {} by A1,CAT_1:84;
  take T/.i;
  thus (T/.i)*(T/.f) = T/.(id a) by A1,A2,Lm1
    .= id(T.a) by Lm2;
end;
