so we're hoping for something like... ??given a differential graded algebroid s and a differential graded s-module w, obtain (??perhaps subject to some sort of finitariness or other sort of niceness condition on s and w?? or something??) a differential graded s-opmodule w* st "weak homming from w* is equivalent to weak tensoring with w" ... ??or something like that...

hmm, not quite sure how well that parses yet... ??...

??so let's try to work out the alleged motivating examples that we think we know...

(s,w) is supposed to be the "weighted diagram scheme" for homotopy cokernel... or something like that... so the dg algebroid s should have two objects d and c, and be free on one degree zero cycle m from d to c... ??and the s-module w should be the strict weight for cokernel and the weak weight for homotopy cokernel ... ??or something like that?? ...