The Toposic Hom Conjecture

Let $K$ be an absolutely finitely generated field, meaning that it is finitely generated over its prime field. Let

$$\mathscr{C}=\mathbf{Sch}_{K}^{\operatorname{ft}}[\operatorname{UH}^{-1}]$$

be the localization of the category of finite type $K$-schemes at the universal homeomorphisms, and let $X,Y\in\operatorname{Ob}(\mathscr{C})$. Write $\operatorname{Shv}(X_{\operatorname{\acute et}})$ and $\operatorname{Shv}(Y_{\operatorname{\acute et}})$ for their étale topoi, and let the superscript $\bullet$ denote the class of admissible morphisms specified in the source. The Toposic Hom Conjecture. The natural map

$$\operatorname{Mor}_{\mathscr{C}}(X,Y)\longrightarrow \operatorname{Mor}_{\operatorname{Shv}(K_{\operatorname{\acute et}})}^\bullet(\operatorname{Shv}(X_{\operatorname{\acute et}}),\operatorname{Shv}(Y_{\operatorname{\acute et}}))$$

is a bijection. This is the paper's general étale reconstruction statement; the supplied material does not state whether it is proved in full generality, so its resolution remains to be checked.