In an abelian category $\mathscr A$ we encounters the notions of kernel, cokernel, chain homology, derived functors, etc. These notions are frequently referred to as functors, and yes, they actually behave functorially.
But lately it comes to my mind that, for example, which object is $H_n(X)$ for $X_\bullet$ a chain complex in $\mathscr A$, exactly? Formally it is the cokernel $\ker\partial_n/\operatorname{im}\partial_{n+1}$; one may realize it with some kernel and cokernel "functors", whose target is the arrow category $\text{Arr}(\mathscr A)$, and applying the functor that takes codomain $\text{Arr}(\mathscr A)\to\mathscr A$ to finally obtain an object in $\mathscr A$. The last functor being unambiguous, it suffices to figure out the precise meaning of kernel and cokernel functors; by duality we focus on the kernel functor.
$\ker$ should be a functor $\text{Arr}(\mathscr A)\to\text{Arr}(\mathscr A)$(e.g. in Section 3 of this nLab page), but I've never seen it defined explicitly. For a morphism $f:A\to B$, which arrow is $\ker f$, among ones that are isomorphic to each other by a unique isomorphism? In $\text{Ab}$ or $\text{$R$-Mod}$ there is a canonical construction, but what about a general abelian category? If we just choose for each $f$ an arbitrary $\ker f$ among all isomorphic ones, wouldn't this involve an ultimately strong form of Axion of Choice?
The same issue arises when we consider the derived category. For example, assuming $\mathscr A$ has enough projectives, we claim that there is a canonical functor $F:\mathscr A\to\mathscr D_{\bullet}(\mathscr A)$, by sending any object $A$ to its projective resolution, which we argue to be well-defined by showing that projective resolution of $A$ is unique up to a unique isomorphism. But which chain is $F(A)$ exactly?
Question: is there a standard way in category theory to resolve this? Do we form a new category by identifying those objects that are isomorphic in some sense? I could not really formalize this idea; when I tried to work it out for $\ker$, it seems that domain functor is no longer well-defined(with target in $\mathscr A$) in the new category, so homology cannot be defined in this way. I feel completely lost.
Thanks for any advice or reference.