$\newcommand{\M}{\mathcal{M}}\newcommand{\ML}{\underline{\mathcal{M}}}\newcommand{\N}{\mathcal{N}}\newcommand{\NL}{\underline{\mathcal{N}}}\newcommand{\V}{\mathscr{V}}\newcommand{\VL}{\underline{\mathscr{V}}}\newcommand{\hom}{\mathsf{hom}}\newcommand{\op}{^{\mathsf{op}}}\require{AMScd}$For a closed monoidal category $\V$, there is a correspondence between closed $\V$-modules and *strongly* tensored (and cotensored) $\V$-enriched categories. My question concerns whether or not we can weakened the hypotheses of this theorem. I put the definitions of all used terms at the end.

Now the correspondence, as far as I'm aware of it/have been able to verify myself, contains the following points:

- Any closed $\V$-module $\M$ can be canonically enriched to a $\V$-category $\ML$
- The module tensor $\odot$ is, in this correspondence, always promotable to a
*strong*tensor making $\ML$ a strongly tensored $\V$-category - Any strongly tensored $\V$-category has a canonical $\V$-module structure given by the same tensor (where the hypothesis of being strongly, as opposed to weakly, tensored is used in a crucial way)
- $\V$-module functors $\M\to\N$ of closed $\V$-modules always lift to enriched $\V$-functors $\ML\to\NL$ under the above correspondences
- Enriched $\V$-functors $\ML\to\NL$ of always descend to $\V$-module functors $\M\to\N$
**if we assume the cotensor exists**i.e. assume the modules are "doubly" closed - Enriched natural transformations are in correspondence between 'module' transformations of module/enriched functors between closed $\V$-modules/strongly tensored $\V$-categories

Only one of these points seemed to require $\pitchfork$ when I checked the details. Moreover, $\pitchfork$ was only required for one little detail and was used exactly once in exactly one diagram. This makes me wonder if we can state the correspondence without referencing "doubly" closed modules at all. My question is whether or not this is true.

Specifically, here was the thing I required $\pitchfork$ to show:

We take a *doubly* closed $\V$-module $\M$ and a closed $\V$-module $\N$. We suppose $F:\ML\to\NL$ is a $\V$-enriched functor, and want to make it a $\V$-module functor. We define $\alpha^F:v\odot Fm\to F(v\odot m)$ as the $\hom_\N(Fm,-)$ adjunct of: $$v\overset{\eta_v}{\longrightarrow}\hom_\M(m,v\odot m)\overset{F_{m,v\odot m}}{\longrightarrow}\hom_\N(Fm,F(v\odot m))$$Where $\eta$ denotes the relevant adjunction unit.

If we want $(F;\alpha^F)$ to define a module functor we want $\alpha^F$ to be natural in both variables. It suffices to show naturality in each variable separately; I was able to show that $\alpha^F$ is natural in the first variable $v$, but in order to show it is natural in $m$ I required $\pitchfork$: I required the following diagram to always commute in $\M$: $$\begin{CD}v@>\eta^{m'}_v>>\hom_\M(m',v\odot m')\\@V\eta^m_vVV@VV\hom_\M(f,1)V\\\hom_\M(m,v\odot m)@>>\hom_\M(1,1\odot f)>\hom_\M(m,v\odot m')\end{CD}$$If the two variable adjunction involving $\pitchfork$ exists, this is possible to show. However, I could not show this commutes without that assumption. Question: *is it possible to show the diagram commutes without using $\pitchfork$*?

I hope this is true, because it is mildly annoying that we need $\pitchfork$ once and only once. Alternatively, I'd be interested if someone had a good proof that $\alpha^F$ is natural that doesn't require the above diagram to commute.

Definitions:

I use definition $6$ of this for '$\V$-module'.

Here, a "doubly" closed $\V$-module is a $\V$-module $\M$ equipped with a two variable adjunction: $$\M(v\odot m,n)\cong\V(v,\hom(m,n))\cong\M(m,v\pitchfork n)$$Where $\odot:\V\times\M\to\M,\pitchfork:\V\op\times\M\to\M,\hom:\M\op\times\M\to\V$ are functors. $\odot$ is referred to as the tensor and $\pitchfork$ as the cotensor. A closed $\V$-module is the same, just without the data of $\pitchfork$.

A *strongly* tensored $\V$-enriched category $\underline{\M}$ is a $\V$-enriched category together with a $\V$-functor (in the sense that it is a $\V$-functor in each variable separately) $\odot:\VL\times\ML\to\ML$ and isomorphisms $\kappa:\ML(v\odot m,n)\cong\VL(v,\ML(m,n))$ which are **enriched** natural transformations in $m$ and $n$. That this isomorphism is enrichable distinguishes it from what I'd call *weakly* tensored $\V$-categories where $\kappa$ is merely an isomorphism of ordinary functors.

$\V$-module functors are defined in the paper. I made up a definition of "module" transformation myself, but I won't repeat it here as it is not relevant.