Why are functor categories nice?

Question

I was looking at the Yoneda embedding and one motivation is that we are embedding the category into a functor category and "functor categories are nice". What does this mean? What nice properties do functor categories have?

Answer 1

Let $\mathcal{C}$ be a (locally small) category. One good property of the functor category $\mathrm{Fun}(\mathcal{C},\mathrm{Set})$ is that it is complete and cocomplete. In other words arbitrary limits and colimits exists. Since they exist in $\mathrm{Set}$, one can show that in general for any diagram $F$ in $\mathrm{Fun}(\mathcal{C},\mathrm{Set})$ and $X \in \mathcal{C}$ one has

$$(\lim_i F_i)(X)=\lim_i (F_i(X)) \quad \text{and} \quad (\underset{i}{\mathrm{colim}}F_i)(X)=\underset{i}{\mathrm{colim}}(F_i(X)). $$