Functor Category

Let $\mathcal{C},\mathcal{D}$ be a pair of Categories. The functor category $[\mathcal{C},\mathcal{D}]$ is the category where

• Objects are Functors $\mathcal{C}\to\mathcal{D}$
• Morphisms $F\to G$ are Natural Transformations

Properties

• Limits and Colimits in functor categories are computed pointwise.

  Explicitly, consider a diagram $F:\mathcal{J}\to[\mathcal{C},\mathcal{D}]$. This diagram has a limit $\lim_{(j:J)}F(j)$ if and only if we have a limit of the diagram $F:\mathcal{J}\times\mathcal{C}\to\mathcal{D}$ in $\mathcal{D}$, which means that we can compute $(\lim_{(j:J)}F(j))(X)$ as $\lim_{(j:J)}F(j)(X)$.