# The Stacks Project

## Tag 0029

### 4.11. Coequalizers

Definition 4.11.1. Suppose that $X$, $Y$ are objects of a category $\mathcal{C}$ and that $a, b : X \to Y$ are morphisms. We say a morphism $c : Y \to Z$ is a coequalizer for the pair $(a, b)$ if $c \circ a = c \circ b$ and if $(Z, c)$ satisfies the following universal property: For every morphism $t : Y \to W$ in $\mathcal{C}$ such that $t \circ a = t \circ b$ there exists a unique morphism $s : Z \to W$ such that $t = s \circ c$.

As in the case of the pushouts above, coequalizers when they exist are unique up to unique isomorphism, and this follows from the uniqueness of equalizers upon considering the opposite category. 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 1061–1081 (see updates for more information).

\section{Coequalizers}
\label{section-coequalizers}

\begin{definition}
\label{definition-coequalizers}
Suppose that $X$, $Y$ are objects of a category $\mathcal{C}$
and that $a, b : X \to Y$ are morphisms. We say a morphism
$c : Y \to Z$ is a {\it coequalizer} for the pair $(a, b)$ if
$c \circ a = c \circ b$ and if $(Z, c)$ satisfies the following
universal property: For every morphism $t : Y \to W$
in $\mathcal{C}$ such that $t \circ a = t \circ b$ there exists
a unique morphism $s : Z \to W$ such that $t = s \circ c$.
\end{definition}

\noindent
As in the case of the pushouts above, coequalizers when
they exist are unique up to unique isomorphism, and this follows
from the uniqueness of equalizers upon considering the opposite
category. There is a straightforward generalization of this definition
to the case where we have more than $2$ morphisms.

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