so given a dg algebroid s and a dg s-module w and a dg s-opmodule w*, with "weakly homming from w into a dg s-module x" naturally equivalent (??in a nice way?? or something??) to "weakly tensoring x with w*" ... ??what are we really (...) saying here? are we saying that w and w* are adjoint 1-cells in some 2-category?? or something??

well, let's see... if tensoring with w* is just like homming from w, then w tensored with w* had better have a special "point" corresponding to the identity morphism of w ... or something... ??so we're thinking of w as a dg bi-module from the unit dg algebroid to s, and of w* as a dg bi-module from s back to the unit dg algebroid... and the special point that we're talking about is a dg bi-module morphism from the unit dg endo-bi-module of the unit dg algebroid to the composite endo-bi-module w # w* ... ??or something... and this morphism should have the property that... ??what??...

hmm, also, should we have that the unit dg endo-bi-module of s, hommed from w as a dg s-module, gives the dg s-opmodule w*?? or something like that?? ... i feel like i'm doing this very unsystematically... is there some more straightforward way of assembling the facts here, seeing various parallels ... ??or something...

??maybe i should try asking todd about some of this stuff... ??...