# When do powers and ends in functor categories act pointwise?


• Just a courtesy: I am reading an introductory text on categories, where would I find the material to understand this question? – Andrea Feb 8 '16 at 9:49
• @AndreaDiBiagio: if I remember right, all the material needed appears in Mac Lane’s Categories for the working mathematician. On the other hand, the first place I remember starting to understand ends and powers (aka cotensors) over general monoidal categories was from Kelly’s Basic concepts of enriched category theory – Peter LeFanu Lumsdaine Feb 21 '16 at 9:29

Yes, this is correct. It is generally true that the evaluation functors $\text{ev}_Z \colon [\mathcal{J},\mathcal{D}] \to \mathcal{D}$ for $Z \in \mathcal{J}$ jointly create limits (and colimits). So a limit in $\mathcal{C} = [\mathcal{J},\mathcal{D}]$ can be computed pointwise if and only if the corresponding pointwise limits exist in $\mathcal{D}$. In particular when $\mathcal{D}$ is complete, the limit defining $(F \multimap G)X$ in $\mathcal{C} = [\mathcal{J},\mathcal{D}]$ exists and is preserved by evaluation at $Z$.