If $\textbf{C}$ is an abelian category,thenthe category of presheaves of abelian groups on $\textbf{C}$ is abelian.Is $\textbf{C}$ abelian necessary?

Question

I am looking at an exercise which asks that, if, $\textbf{C}$ is abelian, then the category of presheaves of abelian groups on $\textbf{C}$ is abelian. The proof of this proceeds as you would expect, by tediously checking every property to hold in the presheaves category. But it seems to me you never have to use the abelian-ness of $\textbf{C}$, only the abelian ness of $\textbf{Ab}$. For instance, given natural transformations $\alpha$, $\beta$: F $\rightarrow$ G, we define an operation on them as $(\alpha + \beta)_X = \alpha_X + \beta_X$ where the operation on the right arises from the fact that $Ab$ is abelian. Similarly, throughout the proof you use the abelianness of $Ab$, and not $C$. Am I missing something, or do we not need $\textbf{C}$ to be abelian?

Answer 1

No, you don't need $\mathbf{C}$ to be abelian. In fact the category $[\mathbf{C},\mathbf{A}]$ is abelian whenever $\mathbf{A}$ is abelian. This is true because abelian categories can be defined entirely in terms of limits and colimits, and the limits/colimits in $[\mathbf{C},\mathbf{A}]$ are constructed pointwisely, from limits/colimits in $\mathbf{A}$.