# Image (category theory)

Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property:

1. There exists a morphism such that .
2. For any object Z with a morphism and a monomorphism such that , there exists a unique morphism such that .

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.

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

## Examples

In the category of sets the image of a morphism is the inclusion from the ordinary image to . 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 can be expressed as follows:

im f = ker coker f

In an abelian category (which is in particular binormal), if f is a monomorphism then f = ker coker f, and so f = im f.