[[Morphism]]
# Regular epimorphism

A **regular epimorphism** is a morphism out of some object $X$ which occurs as the [[Equalizer and coëqualizer|coëqualizer]] of some parallel pair of morphisms into $X$. #m/def/cat 
In particular by the universal property of the coëqualizer it is an [[Morphism|epimorphism]].

> [!check]- Proof
> Let $f,g: Y\to X$ and $q : Q \to X$ be their equalizer.
> Let $a,b : X \to Z$ so that $at = bt := h$.
> Since the universal property demands the factorization of $h$ via $q$ be unique,
> it follows that $a = b$. <span class="QED"/>

See [[Regular monomorphism]] for the dual notion.

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