Does anybody know where to find an English proof of the following proposition
Let $(X, \mathcal{O}_X)$ be a locally ringed space, $Y = \operatorname{Spec} A$ an affine scheme. Then the natural map
$$\begin{align}\operatorname{Hom}(X, Y ) &\to \operatorname{Hom}(A, Γ(X, \mathcal{O}_X)),\\
(f, f^\flat) &\mapsto f^\flat_Y ,\end{align}$$
is a bijection.
This is Prop. 3.4 in Görtz's and Wedhorn's Algebraic Geometry I (there the proof is only given for the case where $X$ is a scheme) and Proposition 1.6.3 in the 1971 edition of Grothendieck's Eléments de Géométrie Algébrique (full proof given there, but it's in French and AFAIK there is no translation).
Best Answer
Here's my effort at a translation. The original text uses $S = Y$, as I will.