Automorphisms of categories of schemes

Abstract: Given two schemes $S$ and $S&#39;$, we prove that every equivalence between $\mathbf{Sch}<em>S$ and $\mathbf{Sch}</em>{S&#39;}$ comes from a unique isomorphism between $S$ and $S&#39;$. This eliminates all Noetherian and finite type hypotheses from a result of Mochizuki and fully answers a programme set out by Brandenburg in a series of questions on MathOverflow in 2011.