Ring Epimorphism Preserves Unity

Theorem
Let $A$ be a ring with unity $1$.

Let $B$ be a ring.

Let $f: A \to B$ be a ring epimorphism.

Then $\map f 1$ is a unity of $B$.

Proof
By definition, $f$ is a semigroup homomorphism between multiplicative semigroups.

A unity of a ring is by definition an identity element of its multiplicative semigroup.

Thus the result follows from Epimorphism Preserves Identity.