1.Context
Let $(C, \otimes, I, a, l,r)$ be a monoidal category. Here $r$ denotes the right unitor.
Suppose $S: C^{op} \xrightarrow{\sim} C$ is an equivalence of categories with inverse $S’$. Assume that there are bijections $\phi_{X,Y,Z}: \operatorname{Hom}_C(X \otimes Y,SZ) \xrightarrow{\sim} \operatorname{Hom}_C(X, S(Y \otimes Z))$ natural in $X,Y,Z$. This makes $C$ a star-autonomous category. Note that we do not assume that $S$ is a monoidal equivalence. For simplicity suppose that the associator $a$ is the identity and that $S$ and $S'$ are strict inverses.
Define the functor $⅋: C \times C \rightarrow C$ as the composite $C \times C \xrightarrow{\tau}C \times C \xrightarrow{(S',S')} C^{op} \times C^{op}= (C \times C)^{op}\xrightarrow{\otimes^{op}} C^{op}\xrightarrow{S} C$. Here $\tau$ denotes the functor that switches components, while $\otimes^{op}$ is the opposite functor of the monoidal product $\otimes$.
2.Question
Does the equality $$\phi_{A ⅋ B,I,S'B \otimes S'A}(r_{A ⅋ B})=\operatorname{id_A} ⅋ \: \phi_{B,I,S'B}(r_B)$$ hold for any two objects $A,B \in C$?
In the monoidal category of finite dimensional vector spaces over a field with usual duality functor (and the choice of $\phi$ arising from the standard choice of coevaluation and evaluation for the rigid structure) the equality holds. This essentially follows from my answer here together with the fact that the left unitor is natural. For any other choice of $\phi$ that arises from a change of coevaluation/evaluation the equality remains true. More precisely, we can scale each coevaluation $\text{coev}_X:k \rightarrow X^* \otimes X$ by a non-zero scalar $\lambda_X$. This scales any $\phi_{-,X,-}$ by $\lambda_X$. Then the equality still holds since both sides of the equation are scaled by $\lambda_I$.
I tried other examples (in particular non-rigid ones) – to no avail. What is a good place to look for counterexamples to the equality? In particular, I realized that I don't know what the natural transformation $\phi_{X,Y,Z}: \operatorname{Hom}_C(X \otimes Y,SZ) \xrightarrow{\sim} \operatorname{Hom}_C(X, S(Y \otimes Z))$ looks like for many non-rigid, non-posetal star-autonomous categories (e.g. the category of sup-lattices, of coherence spaces, of phase spaces). Often I can show that the natural transformation exists, but getting my hands dirty proved rather difficult. Any help in that respect would be appreciated.
