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;
reserve f,g for Morphism of C;

theorem Th83:
  c is_a_coproduct_wrt i1,i2 & h in Hom(c,c) & h(*)i1 = i1 & h(*)i2 =
  i2 implies h = id c
proof
  assume that
A1: cod i1 = c & cod i2 = c and
A2: for d,f,g st f in Hom(dom i1,d) & g in Hom(dom i2,d) ex h st h in
  Hom(c,d) & for k st k in Hom(c,d) holds k(*)i1 = f & k(*)i2 = g iff h = k and
A3: h in Hom(c,c) & h(*)i1 = i1 & h(*)i2 = i2;
  i1 in Hom(dom i1,c) & i2 in Hom(dom i2,c) by A1;
  then consider i such that
  i in Hom(c,c) and
A4: for k st k in Hom(c,c) holds k(*)i1 = i1 & k(*)i2 = i2 iff i = k by A2;
A5: id c in Hom(c,c) by CAT_1:27;
  (id c)(*)i1 = i1 & (id c)(*)i2 = i2 by A1,CAT_1:21;
  hence id c = i by A4,A5
    .= h by A3,A4;
end;
