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The notion of monoidal category is a categorification of the notion of monoid. If $M$ is a monoid, consider the monoidal category $Vec_M$ (of $M$-graded vector spaces over a field $\mathbb{k}$).

If an object of a monoidal category admits a left dual then it is unique up to isomorphism [EGNO, Proposition 2.10.5]. Now, let $M$ be a monoid, and let $m$ be an element admiting left inverses $m'$ and $m''$. Then the objects $\delta_{m'}$ and $\delta_{m''}$ of $Vec_M$ should be left duals of $\delta_{m}$, so should be isomorphic by previous proposition, and then $m'$ and $m''$ should be the same element.

But in a monoid, an element can have two distinct left inverses (see the answers here), which contradicts the previous paragraph. Where is the mistake?

Reference

[EGNO] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.

Sebastien Palcoux
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It is not true that $\delta_{m'}$ is a left dual of $\delta_m$ unless $m'$ is inverse to $m$ on both sides. Indeed, for $\delta_{m'}$ to be left dual to $\delta_m$, you need a map $\delta_{m'}\otimes \delta_m\to 1$ and a map $1\to\delta_m\otimes\delta_{m'}$ that make certain diagrams commute. If $mm'$ and $m'm$ are not both equal to the identity element, then one of these maps will be forced to be $0$, and so the required diagrams will not be able to commute (they will end up saying the identity maps on $\delta_m$ and $\delta_{m'}$ have to be equal to $0$).

Eric Wofsey
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