You shouldn’t expect presheaf quotients to satisfy descent. A good heuristic here is that the inclusion of sheaves into presheaves is a right adjoint, so it will preserve limits but not colimits in general

@Satan'sMinion Any chance you’d be willing to provide some more details?

@LSpice that is correct! Edited

Nice! I had forgotten about this question, but it’s exactly the kind of answer I would have wanted. I think the idea I was missing at the time was the Iwasawa decomposition

Lurie’s book “Higher Topos Theory” is a classic reference for infinity categories, and the book “A Study in Derived Algebraic Geometry” by Gaitsgory-Rozenblyum is a pretty good introduction to DAG the way geometric Langlands theorists typically think about it. That said, there are probably numerous shorter/more user-friendly introductions to both subjects online

Also, some ideas which feature heavily in modern versions of the theory (and are absent from those notes) are higher category theory and derived algebraic geometry. The former is used everywhere (often in the guise of DG categories), and the latter is mostly used on the “spectral” side of things. For example, the derived geometric Satake equivalence of Bezrukavnikov and Finkelberg is an equivalence of two stable infinity categories, one of which is (Ind)-coherent sheaves on a derived scheme

That’s the one!

Edward Frenkel has some introductory notes on conformal field theory and the Langlands program which are pretty good. You could start there and learn things as you need them. Since the field is pretty massive where you should go from there depends on your interests

ncatlab.org/nlab/show/paracategory ?

“Natural” morphisms $X \to Y$ in a category usually come from applying a natural transformation, hence the name. I usually just use the term “standard” (e.g. the standard basis) when something isn’t canonical but feels intuitive.

In general, if you have two nontrivial, connected, and nonintersecting open subsets $U$ and $V$ in some connected $X$ then the constant sheaf $\mathbb{Z}_X$ on $X$ can’t be flasque. This is because locally constant functions on the disjoint union of $U$ and $V$ are isomorphic to $\mathbb{Z}^2$