A bijection of hom-sets that does not exhibit a cotensor

In the witness of [L1], the sole hom-category is the one-object category $E$ with endomorphism monoid $\mathbb{N}$, and the weight $X$ is the discrete two-object category. The candidate is the sole enriched object. Its underlying hom-set and the underlying set of $[X,E]$ are both singletons, so the required underlying bijection is present and is natural in the only test object.

The enriched cotensor property would require $E\cong [X,E]\cong E\times E$. This is impossible because the endomorphism monoids $\mathbb{N}$ and ${\mathbb{N}}^2$ are not isomorphic. Thus the candidate has the expected natural underlying hom-set bijection but is not a cotensor.