Going from Closed Categories to Monoidal Categories

Question

EDIT: I trimmed down the exposition a bit. I really just wanted everyone to know what my approach has been, but what I had was a bit bloated.

Suppose we have a closed category $V$ as defined here, and suppose that for each $b \in V$, $[b, -]$ has a left adjoint $-\otimes b$. My conjecture is that $- \otimes -$ makes $V$ into a monoidal category. To this end, I've been able to define all the structural natural isomorphisms for a monoidal category (and show that they really are natural isomorphisms) except $\alpha^{-1} \colon a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c$.

My usual approach has been to work with Hom-sets and use adjointness to get to a Hom-set that has a morphism that I've already defined. This approach failed for $\alpha^{-1}$. Instead I was able to show that I'd be able to get to $\alpha^{-1}$ if I had either $$ \Psi^* \colon [a, [b,c]] \rightarrow [a \otimes b, c] $$ or $$ T \colon [a, b] \rightarrow [a \otimes c, b \otimes c] $$

Question: Given a closed catgory $V$ such that $[b, -]$ has a left adjoint $- \otimes b$ for each object $b \in V$, how can we (or can we) define a natural map $\alpha^{-1} \colon a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c$?

If not, what is a counterexample to this conjecture? It seems unlikely to me that we would be able to define every structural morphism for a monoidal category except $\alpha^{-1}$.

Answer 1

According to G.M. Kelly's Examples of Non-monadic Structures on Categories (page 63), downloadable here, $\alpha$ is not necessarily an isomorphism if we only have a closed category with an adjoint to $[-,-]$. Unfortunately, as far I can tell, Kelly does not give a counterexample.

Thanks to Buschi Sergio in the comments for this question for pointing me to Day and LaPlaza's paper On Embedding Closed Categories which has an example of this behavior. Once I understand this example better, I'll try to explain it here. It uses a certain subcategory of the category of topological spaces to get the desired behavior.

Regardless, if we require the adjoint to be "internal" in the sense that we have a natural isomorphism $$ \Psi^* \colon [a, [b, c]] \rightarrow [a \otimes b, c] $$

then everything works out and $\alpha$ is an isomophism, and the resulting structure is a monoidal category.