Kerodon

Example 2.1.6.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories which admit finite products, endowed with the cartesian monoidal structure described in Example 2.1.3.2. For any functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, we can regard the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ as endowed with the lax monoidal structure described in Example 2.1.5.17. This lax monoidal structure is a monoidal structure if and only if the functor $F$ preserves finite products. If this condition is satisfied, then the original functor $F$ inherits a monoidal structure (Remark 2.1.6.12).