Proposition 1.4.4 in F. Borceux Handbook of Categorical Algebra II says that if $\sf{A}$ is an abelian category and $\sf{C}$ is a small category, then the category of functors $\operatorname{Fun}(\sf{C},\sf{A})$ is abelian.
I have two questions:
- Is this true without demanding $\sf{C}$ to be small? The only place where Borceux proof seem to demand this condition is where he uses that in this case limits and colimits are computed pointwise. But I think I know how to prove that products, kernels and cokernels are computed pointwise without demanding $\sf{C}$ to be small.
- If the theorem is true without this condition, then why aren't limits and colimits computed pointwise in functor categories without this assumption?
Best Answer
If $\mathcal{C}$ is not small, the functor category $\operatorname{Fun}(\mathcal{C},\mathcal{A})$ will not be locally small in general. The morphisms in this category are natural transformations, whose components are indexed in $\operatorname{Ob}\mathcal{C}$, which is a proper class/large set if $\mathcal{C}$ is not small. Thus $\operatorname{Fun}(\mathcal{C},\mathcal{A})$ cannot be $\mathsf{Ab}$-enriched and hence cannot be abelian.