Kerodon

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Remark 8.1.5.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $X$ and $Y$. For every category $\operatorname{\mathcal{A}}$, we can use Theorem 2.3.4.1 to identify strictly unitary lax functors $U: [1] \times \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{C}}$ with morphisms of simplicial sets $G: \Delta ^1 \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{A}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. Consequently, Theorem 8.1.5.4 supplies a bijection

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors F: \operatorname{Tw}(\operatorname{\mathcal{A}}) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)} \} \ar [d]^{\sim } \\ \{ \textnormal{Morphisms of simplicial sets \operatorname{N}_{\bullet }(\operatorname{\mathcal{A}}) \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y)} \} . }$