Isomorphism (Category Theory) is Monic

Theorem

Let $\mathbf C$ be a metacategory.

Let $f: C \to D$ be an isomorphism.

Then $f: C \rightarrowtail D$ is monic.

Proof

Since $f$ is an isomorphism, it is a fortiori a split monomorphism.

The result follows from Split Monomorphism is Monic.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.1$: Proposition $2.6$