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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. Regardless, if we require the adjoint to be "internal" in the sense that we ...


2

Of course not. Consider the initial monoidal category $\{1\}$. Every category is then a $\{1\}$-module category. The simplest example of a category which admits no monoidal structure is the empty category.



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