The Grassmannian functor $\mathrm{Gr}_{n, r}$ sends a ring $A$ to the set of rank $n$ summands of the free module $A^{n + r}$. This is a local functor and I.1.3.13 of Demazure and Gabriels book "Introduction to Algebraic Geometry and Algebraic Groups" supposedly contains a proof of this fact.
I've been staring at this proof for a week and I just can't parse it. I would love if there was a second source that I could compare with but everything I've found deals with the Grassmannian as a locally ringed space, not a functor. Does anyone know of a second source that deals with the Grassmannian as a functor?