Suppose $X$ is a smooth algebraic variety (say, in characteristic $0$). It's a folklore result that $D^b\text{Coh}(X)$ is equivalent to the derived category of complexes of sheaves of $\mathcal{O}_X$-modules whose cohomology is coherent. The only reference I could find are these notes, which are written in some exotic language I can't parse. Does anyone know a more canonical reference?

Maybe a more general question is "what is a general reference to cite when using equivalences between classical (Grothendieck-era) derived categories and their more modern analogues?"

(Lurie's Higher Algebra will sometimes mention such equivalences without proving them).