Kerodon

Example 2.1.5.17. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories which admit finite products, and regard $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ as endowed with the cartesian monoidal structures described in Example 2.1.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be any functor, and let $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ be the induced functor of opposite categories. Then the functor $F^{\operatorname{op}}$ admits a lax monoidal structure, which associates to each pair of objects $X,Y \in \operatorname{\mathcal{C}}$ the canonical map $\mu _{X,Y}: F(X \times Y) \rightarrow F(X) \times F(Y)$ in the category $\operatorname{\mathcal{D}}$ (which we can view as a morphism from $F^{\operatorname{op}}(X) \otimes F^{\operatorname{op}}(Y) \rightarrow F^{\operatorname{op}}(X \otimes Y)$ in the category $\operatorname{\mathcal{D}}^{\operatorname{op}}$). The unit for $F$ is given by the unique morphism $\epsilon : F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \rightarrow \mathbf{1}_{\operatorname{\mathcal{D}}}$ in the category $\operatorname{\mathcal{D}}$ (where $\mathbf{1}_{\operatorname{\mathcal{C}}}$ and $\mathbf{1}_{\operatorname{\mathcal{D}}}$ are final objects of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively).