## Taking limits is functorial

It turns out that in that happy situation where a category has -limits for some small category , the assignment of a limit to a functor is itself a functor from to . That is, given functors and from to and a natural transformation we get an arrow , 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.

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## 2 Comments »

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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 |

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 |