So I have (again) a question regarding sheaves and their properties. Specifically, I have a problem with understanding a certain statement of the following proof:
Just for completeness, I added the full proof. I dont understand how to take preimages of maps of sheaves and why we are able to refine that cover even further. Thanks for your help.
I am not sure if I understand your questions. I think you may want find a cover of $S$? We can just give a topological cover of $X$ first, and for any cover $X_i$, we get a topological preimage $f^-1(X_i)$ and find the cover $S_j$. And we give $S_j$ the open subscheme structure, and $S_j$ will factor through $X_i$ (as topological map), finally by Exercise 2.2.12 of Qing Liu's 'Algebraic Geometry and Arithmetic Curve', we know this can become a ringed space morphism.