Epimorphism between unital algebras is unital

Question

How could you prove the following statement?

Let A and B are unital algebras. If $f:A\to B$ is an epimorphism, then $f$ is unital; i.e. $f(1)=1$.

Answer 1

For every $b\in B$ there is $a\in A:f(a)=b$, so $f(1)b=f(1)f(a)=f(1a)=f(a)=b=bf(1)$. So $f(1)$ is a (hence the) identity element of $B$.