Remark 2.1.2.2. Let $\operatorname{\mathcal{C}}$ be a nonunital strict monoidal category. We will say that an object $\mathbf{1} \in \operatorname{\mathcal{C}}$ is a strict unit if it satisfies condition $(\ast )$ of Definition 2.1.2.1. Note that if such an object exists, then it is uniquely determined: it can be characterized as the unit element of the monoid $\operatorname{Ob}(\operatorname{\mathcal{C}})$.
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