Corollary 4.6.8.20. The functor
\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad B \mapsto \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}[B] )}(x,y) \]
has a left adjoint, given by the functor $A \mapsto \Phi (A)$ of Construction 4.6.8.11.