Example 2.3.4.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories. We say that a lax monoidal functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is strictly unitary if the unit $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$ is an identity morphism of $\operatorname{\mathcal{D}}$. It follows from Theorem 2.3.4.1 and Remark 2.2.4.9 that the formation of classifying simplicial sets induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \text{Strictly unitary lax monoidal functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \ar [d]^{\sim } \\ \{ \text{Maps of simplicial sets $B_{\bullet } \operatorname{\mathcal{C}}\rightarrow B_{\bullet } \operatorname{\mathcal{D}}$} \} . } \]