Lemma 2.31.
Let \(C\) be a category with pullbacks and geometric realizations. Then the functor \(\check{C}_{\bullet}\colon \Ar(C) \to \Fun(\simp\catop,C)\) admits a left adjoint
\[\Fun(\simp\catop,C) \xrightarrow{i_!} \Fun(\simp\catop_+,C) \xrightarrow{j^*} \Fun((\simp_+^{\leq 0})\catop,C) \simeq \Fun([1],C),\]
sending a simplicial object \(X_{\bullet}\) to the map \(X_0 \to \colim_{[n] \in \simp\catop} X_n\).
Proof
We have \(\check{C}_{\bullet}(-) = i^*j_*\). The functor \(i^*\) has a left adjoint given by left Kan extension along \(i\colon \simp\catop \hookrightarrow \simp\catop_+\), and the functor \(j_*\) has a left adjoint given by the restriction functor \(j^*\).