Definition 2.1.2.1. Let $\operatorname{\mathcal{C}}$ be a category. A strict monoidal structure on $\operatorname{\mathcal{C}}$ is a nonunital strict monoidal structure $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ for which there exists an object $\mathbf{1} \in \operatorname{\mathcal{C}}$ satisfying the following condition:
- $(\ast )$
For every object $X \in \operatorname{\mathcal{C}}$, we have $X \otimes \mathbf{1} = X = \mathbf{1} \otimes X$ (as objects of $\operatorname{\mathcal{C}}$). Moreover, for every morphism $f: X \rightarrow X'$ in $\operatorname{\mathcal{C}}$, we have $f \otimes \operatorname{id}_{\mathbf{1}} = f = \operatorname{id}_{ \mathbf{1} } \otimes f$ (as morphisms from $X$ to $X'$).
A strict monoidal category is a pair $(\operatorname{\mathcal{C}}, \otimes )$, where $\operatorname{\mathcal{C}}$ is a category and $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a strict monoidal structure on $\operatorname{\mathcal{C}}$.