Remark 2.1.2.11. It is possible to adopt the following variant of Definition 2.1.2.10:
A monoidal category is a nonunital monoidal category $\operatorname{\mathcal{C}}$ which admits a unit, in the sense of Definition 2.1.2.5.
This is essentially equivalent to Definition 2.1.2.10, since a unit $(\mathbf{1}, \upsilon )$ of $\operatorname{\mathcal{C}}$ is uniquely determined up to unique isomorphism (Proposition 2.1.2.9). However, for our purposes it will be more convenient to adopt the convention that a monoidal structure on a category $\operatorname{\mathcal{C}}$ includes a choice of unit object $\mathbf{1} \in \operatorname{\mathcal{C}}$ and unit constraint $\upsilon : \mathbf{1} \otimes \mathbf{1} \simeq \mathbf{1}$.