$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Let $\operatorname{\mathcal{C}}$ be a category. We will sometimes abuse terminology by identifying a monoidal structure $(\otimes , \alpha , \mathbf{1}, \upsilon )$ with the underlying nonunital monoidal structure $(\otimes , \alpha )$ on $\operatorname{\mathcal{C}}$ (or with the underlying tensor product functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$). This is essentially harmless, by virtue of Remark We will also abuse terminology (in a less harmless way) by identifying a monoidal category $(\operatorname{\mathcal{C}}, \otimes , \alpha , \mathbf{1}, \upsilon )$ with the underlying category $\operatorname{\mathcal{C}}$.