Let's assume we have a category which has all kernels and cokernels. If needed we can also assume the category to abelian.
I am currently struggling to show the "first homomorphism theorem"/"fundamental theorem on homomorphisms" for such categories, i.e. let $\phi: A \to B$ be a morphism, then naturally $\phi$ factors through the $C=coimage(\phi)=coker(ker(\phi))$ and the corresponding morphism $\varphi: C\to B$ is epi if $\phi$ is epi.
The problem is now to show, that $\varphi$ is also mono.
Does anyone know if this holds (and how to show this)/what assumptions on the category we need e.g. preadditive. (for the standard examples of abelian categories i.e. rings, modules, vectorspaces, (abelian) groups this holds)