Kerodon

Remark 2.1.1.8. Let $\operatorname{\mathcal{C}}$ be a category equipped with a nonunital monoidal structure $(\otimes , \alpha )$. We will often abuse terminology by identifying the nonunital monoidal structure $(\otimes , \alpha )$ with the underlying tensor product functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$. If we refer to a functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ as a nonunital monoidal structure on $\operatorname{\mathcal{C}}$, we implicitly assume that $\operatorname{\mathcal{C}}$ has been equipped with associativity constraints $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$ satisfying the pentagon identity of Definition 2.1.1.5. Beware that, in the non-strict case, the associativity constraints are an essential part of the data: it is possible to have inequivalent nonunital monoidal categories $(\operatorname{\mathcal{C}}, \otimes , \alpha )$ and $(\operatorname{\mathcal{C}}', \otimes ', \alpha ')$ with $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}'$ and $\otimes = \otimes '$ (see Example 2.1.3.3).