Let $\mathcal{C}$ be symmetric monoidal closed, with tensor product $- \otimes -$ and internal hom $[-,-]$.
In this case, we know that the tensor-hom adjunction internalizes, and $[X \otimes Y, Z] \cong [X, [Y,Z]]$ as objects in $\mathcal{C}$. Are there adjoint functors $L \dashv R$ from $\mathcal{C} \to \mathcal{C}$ for which this isn't true? That is, for which $[LX, Y] \not \cong [X, RY]$ in $\mathcal{C}$?
The obvious idea is to use yoneda:
$$ \begin{aligned} \mathcal{C}(A, [LX, Y]) &\cong \mathcal{C}(A \otimes LX, Y) \\ &\overset{\star}{\cong} \mathcal{C}(L(A \otimes X), Y) \\ &\cong \mathcal{C}(A \otimes X, RY) \\ &\cong \mathcal{C}(A, [X, RY]) \end{aligned} $$
But there's no reason a left adjoint should preserve tensor products, so I would expect step $\star$ to fail for many functors... Unfortunately, I'm struggling to come up with concrete examples where this fails.
Does anybody happen to know any? Obviously I would prefer "natural" examples (in the informal sense), preferably in $R$-mod or similar. Though I suspect the easiest examples will be found in heyting algebras viewed as poset categories.
Thanks in advance ^_^.