Variant 7.1.2.11. Let $\kappa $ be an uncountable cardinal and let $Y$ be a Kan complex which is essentially $\kappa $-small. Suppose we are given a morphism of simplicial sets $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, Y)$. If the simplicial set $K$ is essentially $\kappa $-small, then the following conditions are equivalent:
- $(1)$
The morphism $e$ is a weak homotopy equivalence.
- $(2)$
The morphism $e$ exhibits $Y$ as a copower of $\Delta ^0$ by $K$ in the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$.
Beware that, if $K$ is not assumed to be essentially $\kappa $-small, then it is possible for condition $(2)$ to be satisfied while condition $(1)$ is not (Exercise 7.1.2.12).