Let $\mathcal{C}$ and $\mathcal{D}$ be categories, where $\mathcal{C}$ is an abelian category. We want to say that $\mathcal{C}^\mathcal{D}$ is also an abelian category. However, if $\mathcal{C}$ and $\mathcal{D}$ is big enough, $\hom(F, G)$ in $\mathcal{C}^\mathcal{D}$ is too big to be a set, and hence, we cannot really say it has the structure of an abelian category (at least, in the usual sense).

So, my question is: what are the ways to fix it? I am aware of the option of using universes, but is there any other way(s)?