Proof

Part 1 follows from the fact that f is a bijection. Remains the second part.

Suppose M and M' are two monoids and f is an isomorphism from M to M'.

Since f is an isomorphism, f(e) is the identity e' of M'. Thus f^-1(e') = f^-1( f(e)) = e, the identity of M.

Now suppose a' and b' are elements in M'. Since f is a bijection there exist unique elements a and b in M with a' = f(a) and b' = f(b). Then f(a * b) = a' * b'. Thus we also have that

f^-1(a' * b') = a * b = f^-1(a') * f^-1(b').

And indeed we have shown that f^-1 is also an isomorphism.