# The Stacks Project

## Tag 0027

### 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 product 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.

The code snippet corresponding to this tag is a part of the file categories.tex and is located in lines 1041–1060 (see updates for more information).

\section{Equalizers}
\label{section-equalizers}

\begin{definition}
\label{definition-equalizers}
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 {\it 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 $t = e \circ s$.
\end{definition}

\noindent
As in the case of the fibre product 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.

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