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
 for f being Morphism of c,d holds
  c is_a_coproduct_wrt i1,i2 & dom f = c & cod f = d & f is invertible
  implies d is_a_coproduct_wrt f(*)i1,f(*)i2
proof let f be Morphism of c,d;
  assume that
A1: c is_a_coproduct_wrt i1,i2 and
A2: dom f = c & cod f = d & f is invertible;
  c is_a_coproduct_wrt (0,{0})-->(i1,i2) by A1,Th79;
  then d is_a_coproduct_wrt f*((0,{0})-->(i1,i2)) by A2,Th74;
  then d is_a_coproduct_wrt (0,{0})-->(f(*)i1,f(*)i2) by Th15;
  hence thesis by Th79;
end;
