Kerodon

Example 2.1.2.14. Let $\operatorname{\mathcal{C}}$ be a category. Then every strict monoidal structure $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 2.1.2.1) can be promoted to a monoidal structure $(\otimes , \alpha , \mathbf{1}, \upsilon )$ on $\operatorname{\mathcal{C}}$, by taking $\mathbf{1}$ to be the strict unit of $\operatorname{\mathcal{C}}$ and the associativity and unit constraints to be identity morphisms of $\operatorname{\mathcal{C}}$. Conversely, if $\operatorname{\mathcal{C}}$ is equipped with a monoidal structure $(\otimes , \alpha , \mathbf{1}, \upsilon )$ for which the associativity and unit constraints are identity morphisms, then $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a strict monoidal structure on $\operatorname{\mathcal{C}}$ and $\mathbf{1}$ is the strict unit.