TITLE: Quasi-coherent sheaves in the Functor-of-points approach
QUESTION [14 upvotes]: How do we define quasi-coherent sheaves on schemes?
Say we start by defining the category of affine schemes Aff as CRing$^{op}$ (the opposite category of unitary commutative rings).
In this context we have an obvious way to define quasi-coherent sheaves:
A quasi-coherent sheaf on an affine scheme X=Spec A is just an A-module.
If we now define schemes as presheaves on Aff (satisfying some condition), how do we define what a quasi-coherent sheaf is?
The same question applies also to the operations of pushforward and pullback, which in Aff have obvious definitions.

REPLY [2 votes]: For the non-experts out there, I've found this great expository article by Gomez
http://arxiv.org/PS_cache/math/pdf/9911/9911199v1.pdf
Quasi-coherent sheaves are defined in Definition 2.45 (the same way Dinakar and Yuhao defined them)