reserve C for CatStr;
reserve f,g for Morphism of C;
reserve C for non void non empty CatStr,
  f,g for Morphism of C,
  a,b,c,d for Object of C;
reserve o,m for set;
reserve B,C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,f1,f2,g,g1,g2 for Morphism of C;
reserve f,f1,f2 for Morphism of a,b;
reserve f9 for Morphism of b,a;
reserve g for Morphism of b,c;
reserve h,h1,h2 for Morphism of c,d;

theorem
  a is initial & b is initial implies a,b are_isomorphic
proof
  assume that
A1: a is initial and
A2: b is initial;
  set g = the Morphism of b,a;
  set f = the Morphism of a,b;
A3: Hom(a,b) <> {} by A1;
  take f;
  now
    thus Hom(b,a) <> {} by A2;
    take g;
    thus f*g = id b & g*f = id a by A1,A2,Th45;
  end;
  hence thesis by A3;
end;
