Kerodon

Definition 2.1.2.10. Let $\operatorname{\mathcal{C}}$ be a category. A monoidal structure on $\operatorname{\mathcal{C}}$ is a nonunital monoidal structure $(\otimes , \alpha )$ on $\operatorname{\mathcal{C}}$ (Definition 2.1.1.5) together with a choice of unit $(\mathbf{1}, \upsilon )$ (in the sense of Definition 2.1.2.5). A monoidal category is a category $\operatorname{\mathcal{C}}$ together with a monoidal structure $( \otimes , \alpha , \mathbf{1}, \upsilon )$ on $\operatorname{\mathcal{C}}$. In this case, we refer to $\mathbf{1}$ as the unit object of $\operatorname{\mathcal{C}}$ and the isomorphism $\upsilon : \mathbf{1} \otimes \mathbf{1} \xrightarrow {\sim } \mathbf{1}$ as the unit constraint of $\operatorname{\mathcal{C}}$.