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 iff for b ex f being Morphism of a,b st Hom(a,b) = {f}
proof
  thus a is initial implies for b ex f being Morphism of a,b st Hom(a,b) = {f}
  proof
    assume
A1: a is initial;
    let b;
    consider f being Morphism of a,b such that
A2: for g being Morphism of a,b holds f = g by A1;
    take f;
    thus thesis by A2,Th7,A1;
  end;
  assume
A3: for b ex f being Morphism of a,b st Hom(a,b) = {f};
  let b;
  consider f being Morphism of a,b such that
A4: Hom(a,b) = {f} by A3;
  thus Hom(a,b) <> {} by A4;
  take f;
  thus thesis by A4,Th6;
end;
