Kerodon

Example 2.1.1.7. Let $\operatorname{\mathcal{C}}$ be a category equipped with a nonunital strict monoidal structure $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (in the sense of Definition 2.1.1.1). Then $\otimes $ determines a nonunital monoidal structure on $\operatorname{\mathcal{C}}$ (in the sense of Definition 2.1.1.5) by taking the associativity constraints $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$ to be identity morphisms. Conversely, if $\operatorname{\mathcal{C}}$ is equipped with a nonunital monoidal structure $(\otimes , \alpha )$ where each of the associativity constraints $\alpha _{X,Y,Z}$ is an identity morphism, then $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is a nonunital strict monoidal structure on $\operatorname{\mathcal{C}}$.