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Let $C$ have all limits of diagrams indexed by $J$, and for each $b\in B$ let $E_b:C^B \rightarrow C$ the evaluation functor that evaluates at $b$.

Given a diagram $F: J \rightarrow C^B$, we can calculate its limit by doing it pointwise, i.e. since for each $b\in B$, $E_b \circ F : J \rightarrow C$ has a limit $v_b$ we may construct a functor in $V: B \rightarrow C$ that's a limit of $F$ by letting $V(b) = v_b$ on objects $b\in B$ and blah blah for arrows.

My question is: if $F$ has a limit, does each $E_b\circ F$ have a limit?

This is similar to the problem of mono/epimorphisms, answered in

Can epi/mono for natural transformations be checked pointwise?

So basically if some property is pointwise true in the original category $C$ then it's true in any functor category $C^B$, but which properties are true the other way around?

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up vote 8 down vote accepted

What you're asking is whether every limit in a functor category $[B,C]$ is a pointwise limit. The answer is yes if C is complete, but not always otherwise. Kelly gives an example in Basic Concepts of Enriched Category Theory, section 3.3, of a limit in a functor category that is not a pointwise limit.

I don't know of any striking examples of always-pointwise properties. One obvious example is that of being an absolute limit or colimit (i.e. one preserved by every functor). The latter are characterized by the fact that a category is Cauchy complete if and only if it has all absolute colimits.

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