Lemma 4.13.3. Let $\mathcal{C}$ be a category, and let $f : X \to Y$ be a morphism of $\mathcal{C}$. Then

1. $f$ is a monomorphism if and only if $X$ is the fibre product $X \times _ Y X$, and

2. $f$ is an epimorphism if and only if $Y$ is the pushout $Y \amalg _ X Y$.

Proof. Omitted. $\square$

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