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