The Stacks project

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

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

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

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

Proof. Assume (1). Choose $X'$ such that $L(X') = \mathbf{1}$, i.e., $X \otimes X' \cong \mathbf{1}$. Denote $L'$ and $R'$ the functors corresponding to $X'$. The equation $X \otimes X' \cong \mathbf{1}$ implies $L \circ L' \cong \text{id}$. Thus $L'$ must be the quasi-inverse to $L$ (which exists by assumption). Hence $L' \circ L \cong \text{id}$. Hence $X' \otimes X \cong \mathbf{1}$. Thus (3) holds.

The proof of (2) $\Rightarrow $ (3) is dual to what we just said.

Assume (3). Then it is clear that $L'$ and $L$ are quasi-inverse to each other and it is clear that $R'$ and $R$ are quasi-inverse to each other. Thus (1) and (2) hold. $\square$

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