In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.

Given a category ${\displaystyle C}$ and a morphism ${\displaystyle f\colon X\to Y}$ in ${\displaystyle C}$, the image of ${\displaystyle f}$ is a monomorphism ${\displaystyle m\colon I\to Y}$ satisfying the following universal property:

Remarks:

The image of ${\displaystyle f}$ is often denoted by ${\displaystyle {\text{Im}}f}$ or ${\displaystyle {\text{Im}}(f)}$.

Proposition: If ${\displaystyle C}$ has all equalizers then the ${\displaystyle e}$ in the factorization ${\displaystyle f=m\,e}$ of (1) is an epimorphism.

Let ${\displaystyle \alpha ,\,\beta }$ be such that ${\displaystyle \alpha \,e=\beta \,e}$, one needs to show that ${\displaystyle \alpha =\beta }$. Since the equalizer of ${\displaystyle (\alpha ,\beta )}$ exists, ${\displaystyle e}$ factorizes as ${\displaystyle e=q\,e'}$ with ${\displaystyle q}$ monic. But then ${\displaystyle f=(m\,q)\,e'}$ is a factorization of ${\displaystyle f}$ with ${\displaystyle (m\,q)}$ monomorphism. Hence by the universal property of the image there exists a unique arrow ${\displaystyle v:I\to Eq_{\alpha ,\beta }}$ such that ${\displaystyle m=m\,q\,v}$ and since ${\displaystyle m}$ is monic ${\displaystyle {\text{id}}_{I}=q\,v}$. Furthermore, one has ${\displaystyle m\,q=(mqv)\,q}$ and by the monomorphism property of ${\displaystyle mq}$ one obtains ${\displaystyle {\text{id}}_{Eq_{\alpha ,\beta }}=v\,q}$.

This means that ${\displaystyle I\equiv Eq_{\alpha ,\beta }}$ and thus that ${\displaystyle {\text{id}}_{I}=q\,v}$ equalizes ${\displaystyle (\alpha ,\beta )}$, whence ${\displaystyle \alpha =\beta }$.

In a category ${\displaystyle C}$ with all finite limits and colimits, the image is defined as the equalizer ${\displaystyle (Im,m)}$ of the so-called cokernel pair ${\displaystyle (Y\sqcup _{X}Y,i_{1},i_{2})}$, which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms ${\displaystyle i_{1},i_{2}:Y\to Y\sqcup _{X}Y}$, on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.