# Kerodon

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Remark 2.1.5.12 (Adjoint Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and suppose we are given a pair of adjoint functors $\xymatrix@1{\operatorname{\mathcal{C}} \ar@ <.4ex>[r]^-{F} & \operatorname{\mathcal{D}} \ar@ <.4ex>[l]^-{G}}$, given by an isomorphism of oriented fiber products $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ (see Notation 2.1.4.19). Applying Corollary 2.1.5.10 (and the dual characterization of colax monoidal functors), we see that the following are equivalent:

• The datum of a lax monoidal structure on the functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

• The datum of a colax monoidal structure on the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

• The datum of a monoidal structure on the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ which is compatible with the monoidal structures on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$.