Exercise 7.1.2.12. Let $\kappa $ be an uncountable cardinal and let $K$ be a connected Kan complex satisfying the following conditions:
The fundamental group $\pi _{1}(K)$ is a simple group of cardinality $\geq \kappa $.
The Kan complex $K$ is acyclic: that is, the homology groups $\mathrm{H}_{n}(K; \operatorname{\mathbf{Z}})$ vanish for $n > 0$.
Show that the projection map $K \twoheadrightarrow \Delta ^0 \simeq \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \Delta ^0)$ exhibits $\Delta ^0$ as a copower of itself by $K$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$. That is, $\Delta ^0$ is a colimit of the constant diagram $\underline{\Delta ^0}_{K}$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$, which is not preserved by the inclusion functor $\operatorname{\mathcal{S}}^{< \kappa } \hookrightarrow \operatorname{\mathcal{S}}^{< \lambda }$ for $\lambda \gg \kappa $.