From top to topos

Let \(f\colon X \to Y\) be a continuous map of topological spaces. Then the preimage functor \(f^{-1}\colon \Open(Y) \to \Open(X)\) preserves finite limits and preserves coverings, hence is a continuous morphism of sites. This gives rise to a functor \(\Open(-)\colon \Top\catop \to \Site^{\cont}\). Composing it with the sheaf functor \((\Site^{\cont})\catop \to \Topos\), we obtain a functor