# Image (category theory)

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

1. There exists a morphism ${\displaystyle g\colon X\to I}$ such that ${\displaystyle f=hg}$.
2. For any object Z with a morphism ${\displaystyle k\colon X\to Z}$ and a monomorphism ${\displaystyle l\colon Z\to Y}$ such that ${\displaystyle f=lk}$, there exists a unique morphism ${\displaystyle m\colon I\to Z}$ such that ${\displaystyle h=lm}$.

Remarks:

1. such a factorization does not necessarily exist
2. g is unique by definition of monic (= left invertible, abstraction of injectivity)
3. m is monic.
4. h=lm already implies that m is unique.
5. k=mg

The image of f is often denoted by im f or Im(f).

One can show that a morphism f is monic if and only if f = im f.

## Examples

In the category of sets the image of a morphism ${\displaystyle f\colon X\to Y}$ is the inclusion from the ordinary image ${\displaystyle \{f(x)~|~x\in X\}}$ to ${\displaystyle Y}$. In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets.

In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism ${\displaystyle f}$ can be expressed as follows:

im f = ker coker f

This holds especially in abelian categories.