Let $\mathcal{V}$ be a closed symmetric monoidal category. By definition we have a natural isomorphism of hom-sets $\text{Hom}_{\mathcal{V}}(X\otimes Y,Z) \cong \text{Hom}_{\mathcal{V}}(X,\mathcal{V}(Y,Z))$.
Viewing $\mathcal{V}$ as a $\mathcal{V}$-category one can show that $-\otimes -:\mathcal{V}\times \mathcal{V}\to \mathcal{V}$ and $\mathcal{V}(-,-):\mathcal{V}^{op} \times \mathcal{V} \to \mathcal{V}$ are $\mathcal{V}$-functors.
Is it also true that they form a $\mathcal{V}$-adjunction in general?
Meaning we have a $\mathcal{V}$-natural isomorphism $\Phi_{X,Z}:\mathcal{V}(X\otimes Y,Z) \to \mathcal{V}(X,\mathcal{V}(Y,Z))$ i.e. satisfying $$\require{AMScd} \begin{CD} \mathcal{V}(X\otimes Y,Z) @>{\Phi}>> \mathcal{V}(X,\mathcal{V}(Y,Z))\\ @VVV @VVV \\ \mathcal{V}(X'\otimes Y,Z') @>{\Phi}>> \mathcal{V}(X',\mathcal{V}(Y,Z')) \end{CD}$$ for any morphisms $g:X'\to X$ and $h:Z\to Z'$.