so consider the process of passing from a symmetric monoidal finitely cocomplete algebroid to a symmetric monoidal finitely homotopy-cocomplete dg algebroid (or something like that) by taking chain complexes...

to what extent is this "full and faithful" or something?? ... and so forth...

similarly for just finitely cocomplete algebroid, and so forth... ??relevance of "reflection functors" involving quiver representations here??...