??so... ??have we got an example here of an ag theory with an epi between dualizable objects, with its mate's cokernel object also dualizable, but where the original epi is not its mate's cokernel's mate's cokernel?? ??namely, the category of Z-filtered fp k-modules (with everything in some filtration stage) with the epi from "k born at 0" to "k born at -1"... it's mate's cokernel object is 0, because it's mate is epi too... but that makes it's mates's cokernel's mate a zero map, and the cokernel of that is an isomorphism, whereas the original epi is not iso here... ???or something??

??and then we've also got examples of the sort-of complementary phenomenon... ...??where the mate's cokernel object isn't dualizable... ??right?? ... namely... the filtered module example above is about P^0, whereas with P^1 it seems that the epi's mate's cokernel object isn't dualizable... i think ...

??so ... the condition on an epi between dualizable objects of being "good" does seem to break down into a couple of non-automatic, non-vacuous stages, where the failure can occur at either stage... first, the mate's cokernel object might not be dualizable; then if it is, the original epi might still not be the mate's cokernel's mate's cokernel ... ??...

...assuming that i didn't miscalculate too badly in these examples...