# The Unapologetic Mathematician

## Taking limits is functorial

It turns out that in that happy situation where a category $\mathcal{C}$ has $\mathcal{J}$-limits for some small category $\mathcal{J}$, the assignment of a limit $\varprojlim_\mathcal{J}F$ to a functor $F$ is itself a functor from $\mathcal{C}^\mathcal{J}$ to $\mathcal{C}$. That is, given functors $F$ and $G$ from $\mathcal{C}$ to $\mathcal{J}$ and a natural transformation $\eta:F\rightarrow G$ we get an arrow $\varprojlim_\mathcal{J}\eta:\varprojlim_\mathcal{J}F\rightarrow\varprojlim_\mathcal{J}G$, and composition of natural transformations goes to composition of the limit arrows.

Since the proof of this is so similar to how we established limits in categories of functors, I’ll just refer you back to that and suggest this as practice with those techniques.

June 24, 2007 Posted by | Category theory | 2 Comments