## 4.10 Equalizers

Definition 4.10.1. Suppose that $X$, $Y$ are objects of a category $\mathcal{C}$ and that $a, b : X \to Y$ are morphisms. We say a morphism $e : Z \to X$ is an equalizer for the pair $(a, b)$ if $a \circ e = b \circ e$ and if $(Z, e)$ satisfies the following universal property: For every morphism $t : W \to X$ in $\mathcal{C}$ such that $a \circ t = b \circ t$ there exists a unique morphism $s : W \to Z$ such that $t = e \circ s$.

As in the case of the fibre products above, equalizers when they exist are unique up to unique isomorphism. There is a straightforward generalization of this definition to the case where we have more than $2$ morphisms.

Comment #8139 by Ahmed on

I think a proof of uniqueness is a good idea since in the next section, uniqueness of coequalizers was argued using that of equalizers.

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