so does a half-exact functor between abelian categories nicely induce a htpy-exact fr between the derived (infinity,1)-categories?? or something?? ?and is this what "restoration of exactness" is really about?? or something??

is the (infinity,1)-category of chain complexes of quiver representations of .->. ess the "walking morphism" wrt some hopefully obvious doctrine?? ??or something??...

??given a symmetric monoidal bi-complete (infinity,1)-category (or something...??maybe "stable" or something?? ... ??also something about "negatives of morphisms" or something??...), and given an "environment" of the same type, consider the hom simplicially-enriched groupoid .... ??or something??...

let's consider an example... "[z/2,-]" as a half-exact endofunctor on the abelian category of fp abelian groups... ??...