Let $(\mathcal{V},\otimes,e)$ be a closed symmetric monoidal category and $\underline{\mathcal{A}}$ a tensored $\mathcal{V}$-category. We will write $\mathcal{A}$ for the underlying category of $\underline{\mathcal{A}}$. I was wondering whether the $\mathcal{V}$-natural isomorphism $$\underline{\mathcal{V}}(v,\underline{\mathcal{A}}(a,b))\cong \underline{\mathcal{A}}(v\otimes a ,b)$$ retains enough information to recover the identities and the compositions in $\underline{\mathcal{A}}$. More specifically, I want to show that the identity $e\to \underline{\mathcal{A}}(a,a)$ is the transpose of the isomorphism $e\otimes a\cong a$, and the composition $\underline{\mathcal{A}}(b,c)\otimes \underline{\mathcal{A}}(a,b)\to\underline{\mathcal{A}}(a,c)$ is the transpose of the composition $$\underline{\mathcal{A}}(b,c)\otimes \underline{\mathcal{A}}(a,b)\otimes a \xrightarrow{\operatorname{id}\otimes \epsilon_{a,b}}\underline{\mathcal{A}}(b,c)\otimes b\xrightarrow{\epsilon_{b,c}}c,$$ where $\epsilon_{a,b}$ is the transpose of the identity.
Assuming that the composition map $c_{a,b,c}$ is indeed equal to the above description, I was able to deduce that the identity has to be given by the above description. (I used the induced natural isomorphism $\mathcal{V}(e,\underline{\mathcal{A}}(a,b))\cong\mathcal{A}(e\otimes a,b)$ and the uniqueness of identity.) However, I cannot manage to prove that the $c_{a,b,c}$ is given as above (except in the trivial case $\mathcal{V}=\mathsf{Set}$). Can someone help me? Any help is greatly appreciated. Thanks in advance.