Remark 2.2.4.9 (00F8)

Remark 2.2.4.9. Let $\operatorname{\mathcal{C}}$ be a monoidal category, let $B\operatorname{\mathcal{C}}$ be the $2$-category obtained by delooping $\operatorname{\mathcal{C}}$ (Example 2.2.2.5), and let $X$ denote the unique object of $B\operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{D}}$ be any $2$-category, and let $Y$ be an object of $\operatorname{\mathcal{D}}$. Then the construction of Remark 2.2.4.7 induces bijections

\[ \{ \text{Lax Functors $F: B\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ with $F(X) = Y$} \} \simeq \{ \text{Lax monoidal functors $\operatorname{\mathcal{C}}\rightarrow \underline{\operatorname{End}}_{\operatorname{\mathcal{D}}}(Y)$} \} \]

\[ \{ \text{Functors $F: B\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ with $F(X) = Y$} \} \simeq \{ \text{Monoidal functors $\operatorname{\mathcal{C}}\rightarrow \underline{\operatorname{End}}_{\operatorname{\mathcal{D}}}(Y)$ } \} . \]

Applying this observation in the case where $\operatorname{\mathcal{D}}= B\operatorname{\mathcal{C}}'$ for some other monoidal category $\operatorname{\mathcal{C}}'$, we deduce that (lax) monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{C}}'$ can be identified with (lax) functors of $2$-categories from $B\operatorname{\mathcal{C}}$ to $B\operatorname{\mathcal{C}}'$.