# 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 »

1. Is there a good reference for this result? I can’t seem to find it in my category theory books including Mac Lane. Thanks!

Comment by bfitzpat22 | February 27, 2012 | Reply

2. I don’t recall offhand, though I’d have guessed CWM. If it’s not in there, I may have just recognized it and thought it worth mentioning separately.

Comment by John Armstrong | February 27, 2012 | Reply