# What is the right adjoint of the tensor product in a closed monoidal functor category?

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?

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: $$\begin{split} \left[F,G\right]A & \cong \int_B \mathscr{A}(A,B) \pitchfork [FB,GB] \\ & \cong \int_B [\mathscr{A}(A,B) \otimes FB,GB] \\ & \cong \int_B [ \mathscr{A}(A,B) \otimes I,[FB,GB]] \end{split}$$ 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 $\mathscr{X}$).
• Thank you very much. I also like these formulas more that the one with the product ($\prod$). The $\prod_{I(x, y)}$ only specifies how the functor under the $\int$ maps objects ($y$); it does not say how this functor maps morphisms. – Wolfgang Jeltsch Feb 2 '16 at 18:24
• To me it seems that these formulas are not type correct. I think $\mathscr A(A, B)$ is an object of $\mathscr V$, while $FB$ and $GB$ are objects of $\mathscr X$ (and $\otimes$ lives in $\mathscr X$). Can you please clarify? – Wolfgang Jeltsch Feb 2 '16 at 18:29
• For objects $U$ of $\mathscr{V}$ and $X$ of $\mathscr{X}$, I use $U \otimes X$ to denote the colimit of $X : \mathscr{I} \to \mathscr{X}$ weighted by $U : \mathscr{I} \to \mathscr{X}$, which is called the tensor product. When $\mathscr{V} = \text{Set}$ this is also denoted $U \cdot X$ and called the copower. – Alexander Campbell Feb 2 '16 at 22:38