Setting.
Let $\mathcal{C}$ be a category and $(\mathcal{V},\otimes,I, \multimap)$ a (symmetric) closed monoidal category. Let $F:\mathcal{C}\rightarrow \mathcal{V}$ be a functor and $X\in \mathcal{V}$ an object. Consider the composite functor $$F_X:\mathcal{C}^{\text{op}}\xrightarrow{F^{\text{op}}} \mathcal{V}^{\text{op}}\xrightarrow{?\multimap X}\mathcal{V}.$$
Assume that the following tensor product of functors $$F_X\boxtimes_{\mathcal{C}}F$$ exists.
A vague question.
Can this tensor product (possibly under some additional assumptions) be simplified? Have you come across it somewhere in the literature or your own research? I am mainly interested in the case $\mathcal{C}=\mathcal{V}^{\times n}$ and $F=\otimes^n.$ Here (assuming that $\mathcal{V}$ is strict monoidal) $\otimes^n$ denotes the (n-1)-fold monoidal product $\otimes^n(X_1, \ldots, X_n)=X_1\otimes \ldots \otimes X_n.$
Motivation.
- If $(\mathcal{V},\otimes,I)$ is even a rigid (symmetric) monoidal category, $\mathcal{C}=\mathcal{V}^{\times n}$, $F=\otimes^n$ and $X=I$, then $F_I\boxtimes_{\mathcal{C}}F\cong Z^n(I)$, where $Z:\mathcal{V}\rightarrow \mathcal{V}$ is the so-called central Hopf monad given by the coend $Z(V)=\int^{X\in \mathcal{V}}X^*\otimes V\otimes X.$ For a finite tensor category over a field $k$, this observation essentially allows one to describe its Davydov-Yetter cohomology as a comonad cohomology.
- I want to understand the above tensor product in a less rigid situtation. Namely, assuming that $(\mathcal{V},\otimes,I)$ is more generally a (symmetric) monoidal category with dualizing object $\bot$ (making it a star-autonomous category), I aim to understand the tensor product $F_\bot\boxtimes_{\mathcal{C}}F$, where $\mathcal{C}=\mathcal{V}^{\times n}$ and $F=\otimes^n$ as before.
