Corollary 2.1.5.10. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories, let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor, let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ be the oriented fiber product of Notation 2.1.4.19, and let $U: \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ denote the forgetful functor $(C,D, \eta ) \mapsto (C,D)$. Then the construction $\mu \mapsto \otimes _{\mu }$ of Proposition 2.1.4.21 restricts to a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Lax monoidal structures on $G$} \} \ar [d] \\ {\begin{Bmatrix} \textnormal{Monoidal structures on $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$}
\\ \textnormal{with $U$ strict monoidal}
\end{Bmatrix}} } \]
(see Example 2.1.6.5).