Affine-target adjunction for schemes ($\mathrm{Hom}(X,\mathrm{Spec}A)\simeq \mathrm{Hom}(A,\Gamma (X,{\mathcal{O}}_X))$)

A morphism of schemes into an affine scheme is uniquely determined by its homomorphism on global sections. Conversely, a ring homomorphism $A\to \Gamma (X,{\mathcal{O}}_X)$ restricts on every affine open subscheme of $X$ to a map into $\mathrm{Spec}A$; equality on affine covers of overlaps glues these maps. No quasi-compactness or affineness assumption on $X$ is needed.