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
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?