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Let $C$ be a closed monoidal category, and let $T : C \to C$ be a monad on the underlying category. Let $\sigma$ be a tensorial strength of $T$ and let $\sigma'$ be a cotensorial strength of $T$. A good reference for these notions is the paper Logical relations for monadic types by J. Goubault-Larrecq, S. Lasota, D. Nowak. It is well-known that $\sigma$ corresponds to a natural transformation

$\tau_{A,B} : \underline{\hom}(A,B) \to \underline{\hom}(T(A),T(B))),$

making $T$ a $C$-functor (at least in the case that $C$ is symmetric, see the early papers by Anders Kock). My first question is: Is it also known and written down somewhere that $\sigma'$ corresponds $1:1$ to a natural transformation

$\tau'_{A,B} : T(\underline{\hom}(A,B)) \to \underline{\hom}(A,T(B)),$

such that two certain diagrams commute? In the introduction of Kock's paper Strong functors and monoidal monads this is mentioned without proof.

Recall that $\sigma$ and $\sigma'$ are called compatible (so that $T$ becomes a monoidal monad) if the following diagrams commute.

$$\begin{matrix} A \otimes T(B) \otimes C & \xrightarrow{\sigma} & T(A \otimes B) \otimes C \\\\ \sigma' \downarrow ~ & & ~ \downarrow \sigma' \\\\ A \otimes T(B \otimes C) & \xrightarrow{\sigma} & T(A \otimes B \otimes C)\end{matrix} ~~~~~~ (\heartsuit) $$

$$\begin{matrix} T(A) \otimes T(B) & \xrightarrow{\sigma} & T(T(A) \otimes B) & \xrightarrow{\sigma'} & T^2(A \otimes B) \\\\ \sigma' \downarrow ~ & & & & ~ \downarrow \mu \\\\ T(A \otimes T(B)) & \xrightarrow{\sigma} & T^2(A \otimes B) & \xrightarrow{\mu} & T(A \otimes B) \end{matrix} ~ ~ ~ ~ ~ ~ ~ ~(\lozenge)$$

Question. What are the corresponding compatibility diagrams for $\tau$ and $\tau'$? I guess that $(\lozenge)$ corresponds to the diagram

$$\begin{matrix} T(\underline{\hom}(T(A),T(B))) & \xrightarrow{\tau'} & \underline{\hom}(T(A),T^2(B)) \\\\ \tau \uparrow ~ & & ~ \downarrow \mu \\\\ T(\underline{\hom}(A,B)) & & \underline{\hom}(T(A),T(B)) \\\\ \tau' \downarrow ~ & & ~ \uparrow \mu \\\\ T(\underline{\hom}(A,T(B))) & \xrightarrow{\tau} & \underline{\hom}(T(A),T^2(B)) \end{matrix}$$

But I cannot even guess what $(\heartsuit)$ should correspond to.

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