In this discussion from the categories mailing there is mention of the following result by Robin Houston, supposedly proved in 2006:

Theorem.Let $\mathcal{C}$ be a symmetric closed monoidal category, and let $D$ be an object. If thereexistsa natural isomorphism $f : A \rightarrow (A \multimap D) \multimap D$, then thecanonicalnatural transformation $g : A \rightarrow (A \multimap D) \multimap D$ (coming from the symmetric closed monoidal structure) is an isomorphism (although it may in general be different from $f$).

In the quoted text the term "closed" is missing. Without it, the passage does not seem to make much sense, so I added it in.

I have skimmed through Robin Houston's papers (especially the 2005 paper entitled "Modelling Linear Logic Without Units (Preliminary Results)" seems relevant), but could not find the result in question. Does someone know a reference?