I am starting to study schemes and I am struggling with the practice of restricting to the affine case. I explain a little more: generally when we want to build a morphism between two schemes $X$ and $Y$ we reduce to the case in which $X,Y$ are affine, we build the morphism in this case and then we check that this family of maps glue to a global morphism. The problem I am struggling with is to check the maps I built can be glued. Is there any general reason for which such construction always give us glueable maps? An explicit example is the following: given a scheme $X$ let $X_{red}$ be the reduced scheme and let $f : X \rightarrow Y$ be a morphism of schemes, I want to build a morphism of schemes $f_{red} : X_{red}\rightarrow Y_{red}$ such that $ i_Y \circ f_{red} = f \circ i_X$, where $i_X,i_Y$ are the inclusion morphisms. For every $x \in X$ I can find affine open subset $U_x$ of $x$ and $V_{f(x)}$ of $f(x)$ such that $f(U_x) \subseteq V_{f(x)}$, then here $f = Spec(\phi)$ for some morphism of rings $\phi$, and I set $f_{red}$ on this open subset to be the map corresponding to the morphism of rings induced by $\phi$ on the reductions. However I don’t understand why these maps should coincide on the intersections.
Thank you for the help!