Proposition 5.3.4.20. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Suppose that, for every morphism $u: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the image $\mathscr {F}(u): \mathscr {F}(C) \rightarrow \mathscr {F}(D)$ is a left covering map (Definition 4.2.3.8). Then the morphism $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Construction 5.3.4.11 is an isomorphism.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$ be an $n$-simplex of the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. We identify $\sigma $ with a diagram $C_0 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$, and each $\tau _ i$ with an $i$-simplex of the simplicial set $\mathscr {F}(C_ i)$. We wish to show that there is a unique $n$-simplex $\tau $ of $\mathscr {F}(C_0)$ satisfying $\lambda _{t}(\sigma , \tau ) = ( \sigma , \{ \tau _ i \} _{0 \leq i \leq n} )$. Note that, for this condition to be satisfied, the simplex $\tau $ must be a solution to the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \ar [d] \ar [r] & \mathscr {F}(C_0) \ar [d] \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r]^{ \tau _ n } & \mathscr {F}(C_ n). } \]
Since the inclusion $\{ 0\} \hookrightarrow \Delta ^ n$ is left anodyne (Example 4.3.7.11), our assumption that the right vertical map is a left covering guarantees that this lifting problem has a unique solution $\tau : \Delta ^ n \rightarrow \mathscr {F}(C_0)$ (Corollary 4.2.4.6). This proves uniqueness. To prove existence, write $\lambda _{t}(\sigma , \tau ) = ( \sigma , \{ \tau '_{i} \} _{0 \leq i \leq n} )$. We wish to prove that $\tau _ i = \tau '_ i$ for $0 \leq i \leq n$. For this, we observe that both $\tau _ i$ and $\tau '_ i$ can be viewed as solutions to a common lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \ar [r]^{ \tau _ i(0) } \ar [d] & \mathscr {F}(C_ i) \ar [d] \\ \Delta ^{i} \ar@ {-->}[ur] \ar [r] & \mathscr {F}( C_ n ). } \]
Since the inclusion $\{ 0\} \hookrightarrow \Delta ^ i$ is left anodyne (Example 4.3.7.11) and the right vertical map is a left covering, the solution to this lifting problem is uniquely determined (Corollary 4.2.4.6). $\square$