$\newcommand{\C}{\mathcal C}\newcommand{\I}{\mathcal I}\newcommand{\D}{\mathcal D}\newcommand{\J}{\mathcal J}$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?



When do powers and ends in functor categories act pointwise?
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 $Z$.


