Kerodon

Example 2.1.2.15. Let $\operatorname{\mathcal{C}}$ be a monoidal category and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Assume that $\operatorname{\mathcal{C}}_0$ contains the unit object $\mathbf{1}$ and is closed under the formation of tensor products in $\operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{C}}_0$ inherits the structure of a monoidal category: the underlying nonunital monoidal structure on $\operatorname{\mathcal{C}}_0$ is given by the construction of Remark 2.1.1.9, and the unit $( \mathbf{1}, \upsilon )$ of $\operatorname{\mathcal{C}}_0$ coincides with the unit of $\operatorname{\mathcal{C}}$.