The Stacks project

Lemma 4.43.4. In a monoidal category $\mathcal{C}, \otimes , \phi , \mathbf{1}, 1$ and with notation as in the proof of Lemma 4.43.1 we have

  1. the arrows $1, r_\mathbf {1}, l_\mathbf {1} : \mathbf{1} \otimes \mathbf{1} \to \mathbf{1}$ agree,

  2. the arrows $l_ X \otimes \text{id}_ Y, l_{X \otimes Y} : \mathbf{1} \otimes X \otimes Y \to X \otimes Y$ agree, and

  3. the arrows $\text{id}_ X \otimes r_ Y , r_{X \otimes Y} : X \otimes Y \otimes \mathbf{1} \to X \otimes Y$ agree.

A monoidal category satisfies the assumptions of [Theorem 5.2, associativity].

Proof. We have seen (1) in the proof of Lemma 4.43.1. We have seen in the proof of Lemma 4.43.1 that $l_ X$ and $l_{X \otimes Y}$ are the unique morphisms such that $\text{id}_\mathbf {1} \otimes l_{X \otimes Y} = 1 \otimes \text{id}_{X \otimes Y}$ and $\text{id}_\mathbf {1} \otimes l_ X = 1 \otimes \text{id}_ X$. Part (2) follows immediately. Part (3) is proved in a similar manner. Jointly with the commutativity of (4.43.0.1) and (4.43.0.2) this means the final statement of the lemma holds. $\square$

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