Lemma 4.43.13. In a symmetric monoidal category $\mathcal{C}, \otimes , \phi , \psi , \mathbf{1}, 1$ we have
the arrows $1 \circ \psi , 1 : \mathbf{1} \otimes \mathbf{1} \to \mathbf{1}$ agree,
the arrows $\text{id}_ X \otimes l_ Y, (l_ X \otimes \text{id}) \circ (\psi \otimes \text{id}_ Y): X \otimes \mathbf{1} \otimes Y \to X \otimes Y$ agree,
A symmetric monoidal category satisfies the assumptions of [Theorem 5.1, associativity].
Proof. We may write $\psi = a \otimes \text{id}_\mathbf {1}$ for a unique isomorphism $a : \mathbf{1} \to \mathbf{1}$. Lemma 4.43.2 implies that $a \otimes \text{id}_\mathbf {1} = \text{id}_\mathbf {1} \otimes a$. Functoriality of $\psi $ says that the diagram
commutes. Thus the top arrow is equal to $a \otimes \text{id}_\mathbf {1} \otimes \text{id}_\mathbf {1}$. Thus (4.43.11.1) for $X = Y = Z = \mathbf{1}$ says that $a \otimes \text{id}_\mathbf {1} \otimes \text{id}_\mathbf {1}$ is equal to its own square. Hence $a = \text{id}_\mathbf {1}$. This proves (1).
Part (2) states that $\psi : X \otimes \mathbf{1} \to \mathbf{1} \otimes X$ is the identity, if we identify the source and the target with $X$ in our monoidal category. This follows from the commutativity of (4.43.11.1) for $\mathbf{1}, X, \mathbf{1}$, namely
commutes and we know all but one of the morphisms $\psi $ in this diagram are equal to the identity.
In addition to (1) and (2) the commutativity of the diagrams (4.43.0.1), (4.43.0.2), (4.43.11.1) and the results of Lemma 4.43.4 imply the final statement of the lemma. $\square$
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