Kerodon

Proof. Without loss of generality, we may assume that $\kappa $ is regular (and therefore of cofinality $> \aleph _0$). Let $Y_0 \subseteq X$ be the image of $e$. We will construct $Y$ as the union of an increasing sequence of $\kappa $-small simplicial subsets

Assume that $Y_ n$ has been defined for some $n \geq 0$. Since $X$ is weakly contractible, the Kan complex $\operatorname{Ex}^{\infty }(X)$ is contractible. It follows that the inclusion $\operatorname{Ex}^{\infty }(Y_ n) \hookrightarrow \operatorname{Ex}^{\infty }(X)$ is nullhomotopic: that is, it admits an extension $f_ n: \operatorname{Ex}^{\infty }(Y_ n)^{\triangleright } \rightarrow \operatorname{Ex}^{\infty }(X)$. Since $\operatorname{Ex}^{\infty }(Y_ n)^{\triangleright }$ is $\kappa $-small (Corollary 4.7.4.16), we can choose a $\kappa $-small simplicial $Y_{n+1} \subseteq X$ containing $Y_{n}$ such that $f_ n$ factors through $\operatorname{Ex}^{\infty }( Y_{n+1} )$. To complete the proof, it will suffice to show that the union $Y = \bigcup _{n} Y_{n}$ is weakly contractible, or equivalently that the Kan complex $\operatorname{Ex}^{\infty }(Y)$ is contractible. This follows from Corollary 3.2.4.15, since $\operatorname{Ex}^{\infty }(Y)$ is the colimit of a diagram of nullhomotopic maps