The Stacks project

Proof. We construct the inverse of the map. Let $\varphi : \Gamma (T, \mathcal{O}_ T) \to \Gamma (X, \mathcal{O}_ X)$ be a ring map. Choose a presentation $X = U/R$, see Spaces, Definition 65.9.3. By Schemes, Lemma 26.6.4 the composition

corresponds to a unique morphism of schemes $g : U \to T$. By the same lemma the two compositions $R \to U \to T$ are equal. Hence we obtain a morphism $f : X = U/R \to T$ such that $U \to X \to T$ equals $g$. By construction the diagram

commutes. Hence $f^\sharp $ equals $\varphi $ because $U \to X$ is an étale covering and $\mathcal{O}_ X$ is a sheaf on $X_{\acute{e}tale}$. The uniqueness of $f$ follows from the uniqueness of $g$. $\square$