The nLab says the following about closed monoidal functor categories:



> 
> Let $C$ be a complete closed monoidal category and $I$ any small category. Then the functor category $[I, C]$ is closed monoidal with the pointwise tensor product, $(F \otimes G)(x) = F(x) \otimes G(x)$.
> 
> 
> 


Now I wonder what the right adjoint of $F \otimes {-}$ is. I suppose, that it fulfills the following equation (which is a generalization of the equation for exponentials in functor categories):


$$(F \multimap G)(x) = \int\_{y : I} \prod\_{I(x, y)} F(y) \multimap G(y)$$


Is this correct? And if yes, is there a more standard way of representing the right adjoint?



What is the right adjoint of the tensor product in a closed monoidal functor category?
The formula you give is correct. However, I prefer to use a power/cotensor rather than an indexed product, for while they are equivalent for ordinary categories, only the former gives the correct formula for the general enriched case. So, if $\mathscr{A}$ is a $\mathscr{V}$-category and $\mathscr{X}$ is a closed monoidal $\mathscr{V}$-category, the internal hom in $[\mathscr{A},\mathscr{X}]$ for the pointwise monoidal structure is given by any of the following equivalent formulas:
\begin{equation}
\begin{split}
\left[F,G\right]A & \cong \int\_B \mathscr{A}(A,B) \pitchfork [FB,GB] \\
& \cong \int\_B [\mathscr{A}(A,B) \otimes FB,GB] \\
& \cong \int\_B [ \mathscr{A}(A,B) \otimes I,[FB,GB]]
\end{split}
\end{equation}
when the indicated limits and colimits exist in $\mathscr{X}$ (here $\pitchfork$ and $\otimes$ denote cotensoring and tensoring with objects of $\mathscr{V}$ and $I$ is the unit object of $\mathscr{X}$).


