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

Remark It is possible to adopt the following variant of Definition

  • A monoidal category is a nonunital monoidal category $\operatorname{\mathcal{C}}$ which admits a unit, in the sense of Definition

This is essentially equivalent to Definition, since a unit $(\mathbf{1}, \upsilon )$ of $\operatorname{\mathcal{C}}$ is uniquely determined up to unique isomorphism (Proposition However, for our purposes it will be more convenient to adopt the convention that a monoidal structure on a category $\operatorname{\mathcal{C}}$ includes a choice of unit object $\mathbf{1} \in \operatorname{\mathcal{C}}$ and unit constraint $\upsilon : \mathbf{1} \otimes \mathbf{1} \simeq \mathbf{1}$.