Remark 2.1.4.23 (Adjoint Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories and suppose we are given a pair of adjoint functors $\xymatrix@1{\operatorname{\mathcal{C}} \ar@ <.4ex>[r]^-{F} & \operatorname{\mathcal{D}} \ar@ <.4ex>[l]^-{G}}$, so that we have 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 Proposition 2.1.4.21 (and the dual characterization of nonunital colax monoidal functors), we see that the following are equivalent:
The datum of a nonunital lax monoidal structure on the functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.
The datum of a nonunital colax monoidal structure on the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.
The datum of a nonunital 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 nonunital monoidal structures on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ (meaning that the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is a nonunital strict monoidal functor).