Is this concept of a left-abelian category studied?

Question

A category is abelian if it is preadditive and

1. it has a zero object,
2. it has all binary biproducts,
3. it has all kernels and cokernels, and
4. all monomorphisms and epimorphisms are normal.

Now we introduce the following concept: A category is left-abelian if it is preadditive and

1. it has a zero object,
2. it has all binary biproducts,
3. it has all kernels and all cokernels of kernals, and:
4. Let $f:X\rightarrow Y$ be a morphism in the category. It must factor (uniquely) through its cokernal of kernal $g:X\rightarrow Z$ by the universal properties. Hence we find a unique $h: Y \rightarrow Z$ such that $f = hg$. Now we demand that $h$ is a monomorphism.

Dually one can define the concept of a right-abelian category. One can show that a category is abelian iff it is left- and right-abelian. Are left- or right-abelian categories studied in the literature?

Answer 1

Some notion of this kind exists, see:

Breitsprecher. Lokal endlich präsentierbar Grothendieck-Kategorien. Mitt. Math. Sem. Giessen Heft, 85:1–25, 1970.

Rump. Locally finitely presented categories of sheaves. Journal of Pure and Applied Algebra, 2010, 214.2: 177-186.