Kerodon

Example 2.1.2.16. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. Then every monoidal structure on $\operatorname{\mathcal{D}}$ determines a monoidal structure on the functor category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, whose underlying nonunital monoidal structure is given by the construction of Remark 2.1.1.10 and whose unit object is the constant functor $\operatorname{\mathcal{C}}\rightarrow \{ \mathbf{1} \} \hookrightarrow \operatorname{\mathcal{D}}$ (and whose unit constraint $\upsilon : \mathbf{1} \otimes \mathbf{1} \simeq \mathbf{1}$ is the constant natural transformation induced by the unit constraint of $\operatorname{\mathcal{D}}$).