When do powers and ends in functor categories act pointwise?

Question

$\newcommand{\C}{\mathcal C}\newcommand{\I}{\mathcal I}\newcommand{\D}{\mathcal D}\newcommand{\J}{\mathcal J}$Let $(\C, \otimes, I, \multimap)$ be a complete closed monoidal category and $\I$ a small category. Then the functor category $[\I, \C]$ is closed monoidal with the pointwise tensor product and its right adjoint defined as follows:\begin{align}(F \otimes G)X & = FX \otimes GX\\(F \multimap G)X & = \int_Y \I(X, Y) \pitchfork (FY \multimap GY)\end{align}Now assume that $\C$ is a functor category $[\J, \D]$ where $\D$ is complete and $\J$ is small. We can write the definition of ${\multimap} : [\I, \C]^{\mathrm{op}} \times [\I, \C] \to [\I, \C]$ as follows:\begin{equation}(F \multimap G)XZ = \left(\int_Y \I(X, Y) \pitchfork (FY \multimap GY)\right)Z\end{equation}I was thinking about pushing the application to $Z$ under the end and the power, which would result in the following equation:\begin{equation}(F \multimap G)XZ = \int_Y \I(X, Y) \pitchfork (FY \multimap GY)Z\end{equation}Is this actually correct? What are the conditions for ends and powers acting pointwise?

Answer 1

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 $Z$.