Kerodon

Proof. We proceed as in the proof of Proposition 3.5.6.12. Fix a morphism of simplicial sets $g: A \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$. Using Remark 4.8.4.10 (and Proposition 1.1.4.12), we see that $g|_{ \operatorname{sk}_{n}(A) }$ can be lifted to a morphism of simplicial sets $u: \operatorname{sk}_{n}(A) \rightarrow \operatorname{\mathcal{C}}$. We will show that $u$ can be extended to the $(n+1)$-skeleton of $A$ (and is therefore classified by a morphism $f: A \rightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ satisfying $\theta (f) = g$; see Remark 3.5.4.21). By virtue of Proposition 1.1.4.12, this is equivalent to the assertion that for every $(n+1)$-simplex $\sigma $ of $A$ having restriction $\sigma _0 = \sigma |_{ \operatorname{\partial \Delta }^{n+1} }$, the composition $(g \circ \sigma _0): \operatorname{\partial \Delta }^{n+1} \rightarrow \operatorname{\mathcal{C}}$ can be extended to an $(n+1)$-simplex of $\operatorname{\mathcal{C}}$. Choose a lift of $g(\sigma )$ to an $(n+1)$-simplex of $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$, which we can identify with a diagram $\tau _0: \operatorname{\partial \Delta }^{n+1} \rightarrow \operatorname{\mathcal{C}}$ which admits an extension $\tau : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$. By construction, $g \circ \sigma _0$ and $\tau _0$ coincide after composing with the comparison map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$. Using Proposition 1.1.4.12 again, we see that $g \circ \sigma _0$ and $\tau _0$ are isomorphic relative to $\operatorname{sk}_{n-1}( \Delta ^{n+1} )$. The desired result now follows from Corollary 4.4.5.3. This completes the proof that $\theta $ is surjective.

Now suppose that we are given a pair of morphisms $f_0, f_1: A \rightarrow \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ satisfying $\theta (f_0) = \theta (f_1)$. We wish to show that the associated maps $u_0, u_1: \operatorname{sk}_{n}(A) \rightarrow \operatorname{\mathcal{C}}$ are isomorphic relative to $\operatorname{sk}_{n-1}(A)$ (the converse is immediate from the definitions). Using Remark 4.8.4.10, we deduce that $u_0$ and $u_1$ coincide on $\operatorname{sk}_{n-1}(A)$. By virtue of Proposition 1.1.4.12, we are reduced to showing that for every nondegenerate $n$-simplex $\sigma $ of $A$, the compositions $u_0 \circ \sigma $ and $u_1 \circ \sigma $ are isomorphic relative to $\operatorname{\partial \Delta }^{n}$. This follows from our assumption that the maps $\theta (f_0), \theta (f_1): A \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ coincide on the simplex $\sigma $. $\square$