Lemma 4.43.3. Let $\mathcal{C}$ be a monoidal category. Let $X$ be an object of $\mathcal{C}$. The following are equivalent

the functor $L : Y \mapsto X \otimes Y$ is an equivalence,

the functor $R : Y \mapsto Y \otimes X$ is an equivalence,

there exists an object $X'$ such that $X \otimes X' \cong X' \otimes X \cong \mathbf{1}$.

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