Definition 10.3.0.2. Let $\operatorname{\mathcal{C}}$ be a category which admits pullbacks. We will say that a morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is a regular epimorphism if it exhibits $Y$ as a coequalizer of the pair of projection maps $\pi _0, \pi _1: X \times _{Y} X \rightarrow X$.
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