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 Th74:
  for f being Morphism of a,b
  for F being Injections_family of a,I st a is_a_coproduct_wrt F &
  dom f = a & cod f = b & f is invertible holds b is_a_coproduct_wrt f*F
proof let f be Morphism of a,b;
  let F be Injections_family of a,I such that
A1: a is_a_coproduct_wrt F and
A2: dom f = a and
A3: cod f = b and
A4: f is invertible;
  thus f*F is Injections_family of b,I by A2,A3,Th66;
  let c;
A5: cods F = I-->dom f by A2,Def16;
  let F9 be Injections_family of c,I;
  assume doms(f*F) = doms F9;
  then doms F = doms F9 by A5,Th17;
  then consider h such that
A6: h in Hom(a,c) and
A7: for k st k in Hom(a,c) holds (for x st x in I holds k(*)(F/.x) = F9/.x
  ) iff h = k by A1;
A8: dom h = a by A6,CAT_1:1;
  consider g being Morphism of b,a such that
A9: f*g = id b and
A10: g*f = id a by A4;
A11: Hom(b,a) <> {} & Hom(a,b) <> {} by A4;
  then
A12: dom g = cod f by A3,CAT_1:5;
A13: cod g = dom f by A2,A11,CAT_1:5;
A14: f*g = f(*)g by A11,CAT_1:def 13;
A15: g*f = g(*)f by A11,CAT_1:def 13;
  cod h = c by A6,CAT_1:1;
  then
A16: cod(h(*)g) = c by A2,A13,A8,CAT_1:17;
  take hg = h(*)g;
  dom(h(*)g) = b by A2,A3,A12,A13,A8,CAT_1:17;
  hence hg in Hom(b,c) by A16;
  let k;
  assume
A17: k in Hom(b,c);
  then
A18: dom k = b by CAT_1:1;
A19: cod k = c by A17,CAT_1:1;
  thus (for x st x in I holds k(*)((f*F)/.x) = F9/.x) implies hg = k
  proof
    assume
A20: for x st x in I holds k(*)((f*F)/.x) = F9/.x;
    now
      cod(k(*)f) = c & dom(k(*)f) = a by A2,A3,A19,A18,CAT_1:17;
      hence k(*)f in Hom(a,c);
      let x;
      assume
A21:  x in I;
      then cod(F/.x) = a by Th62;
      hence k(*)f(*)(F/.x) = k(*)(f(*)(F/.x)) by A2,A3,A18,CAT_1:18
        .= k(*)((f*F)/.x) by A21,Def6
        .= F9/.x by A20,A21;
    end;
    then k(*)f(*)g = h(*)g by A7;
    hence hg = k(*)(id b) by A3,A13,A9,A18,A14,CAT_1:18
      .= k by A18,CAT_1:22;
  end;
  assume
A22: hg = k;
  let x;
  assume
A23: x in I;
  then
A24: cod(F/.x) = a by Th62;
  then
A25: cod(f(*)(F/.x)) = b by A2,A3,CAT_1:17;
  thus k(*)((f*F)/.x) = k(*)(f(*)(F/.x)) by A23,Def6
    .= h(*)(g(*)(f(*)(F/.x))) by A2,A3,A12,A13,A8,A22,A25,CAT_1:18
    .= h(*)((id dom f)(*)(F/.x)) by A2,A12,A10,A24,A15,CAT_1:18
    .= h(*)(F/.x) by A2,A24,CAT_1:21
    .= F9/.x by A6,A7,A23;
end;
