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
  a is_a_coproduct_wrt y .--> id a
proof
  set F = y .--> id a;
  cod(id a) = a;
  hence F is Injections_family of a,{y} by Th64;
  let b;
  let F9 be Injections_family of b,{y} such that
A1: doms F = doms F9;
  take h = F9/.y;
A2: y in {y} by TARSKI:def 1;
  then
A3: cod h = b by Th62;
  dom h = (doms F)/.y by A1,A2,Def1
    .= dom(F/.y) by A2,Def1
    .= dom(id a) by A2,Th2
    .= a;
  hence h in Hom(a,b) by A3;
  let k;
  assume k in Hom(a,b);
  then
A4: dom k = a by CAT_1:1;
  thus (for x st x in {y} holds k(*)(F/.x) = F9/.x) implies h = k
  proof
    assume
A5: for x st x in {y} holds k(*)(F/.x) = F9/.x;
    thus k = k(*)(id a) by A4,CAT_1:22
      .= k(*)(F/.y) by A2,Th2
      .= h by A2,A5;
  end;
  assume
A6: h = k;
  let x;
  assume
A7: x in {y};
  hence F9/.x = k by A6,TARSKI:def 1
    .= k(*)(id a) by A4,CAT_1:22
    .= k(*)(F/.x) by A7,Th2;
end;
