Kerodon

Proof. We first prove that $\theta $ is a surjection. Fix a morphism $g: A \rightarrow \pi _{\leq n}(X)$. Using Remark 3.5.6.11 (and Proposition 1.1.4.12), we see that $g|_{ \operatorname{sk}_{n}(A) }$ can be lifted to a morphism of simplicial sets $u: \operatorname{sk}_{n}(A) \rightarrow X$. We will show that $u$ can be extended to the $(n+1)$-skeleton of $A$ (and is therefore classified by a morphism $f: A \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ satisfying $\theta (f) = g$; see Remark 3.5.4.21). By virtue of Proposition 1.1.4.12 and Variant 3.2.4.12, 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 $(g \circ \sigma _0): \operatorname{\partial \Delta }^{n+1} \rightarrow X$ is nullhomotopic. Choose a lift of $g(\sigma )$ to an $(n+1)$-simplex of $\operatorname{cosk}_{n}^{\circ }(X)$, which we identify with a nullhomotopic map $\tau _0: \operatorname{\partial \Delta }^{n+1} \rightarrow X$. By construction, $g \circ \sigma _0$ and $\tau _0$ coincide after composing with the comparison map $v: X \rightarrow \pi _{\leq n}(X)$. Using Proposition 1.1.4.12 again, we see that $g \circ \sigma _0$ and $\tau _0$ are homotopic relative to the $\operatorname{sk}_{n-1}( \Delta ^{n+1} )$, so that $g \circ \sigma _0$ is also nullhomotopic.

Now suppose that we are given a pair of morphisms $f_0, f_1: A \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ satisfying $\theta (f_0) = \theta (f_1)$. We wish to show that the associated maps $u_0, u_1: \operatorname{sk}_{n}(A) \rightarrow X$ are homotopic relative to $\operatorname{sk}_{n-1}(A)$ (the converse is immediate from the definitions). Using Remark 3.5.6.11, 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 homotopic relative to $\operatorname{\partial \Delta }^{n}$. This follows from our assumption that the maps $\theta (f_0), \theta (f_1): A \rightarrow \pi _{\leq n}(X)$ coincide on the simplex $\sigma $. $\square$