Isomorphisms between curve graphs of infinite-type surfaces are geometric

Let $\phi :\mathcal {C}(S)\to \mathcal {C}(S')$ be a simplicial iso\-mor\-phism between curve graphs of infinite-type surfaces. In this paper, we show that, in this situation, $S$ and $S'$ are homeomorphic and $\phi $ is induced by a homeomorphism $h:S\to S'$.