a couple of times recently i've found myself getting confused about the relationship between various concepts of dual... now i'm thinking that there's a very simple relationship between two of them that i've understood in the past and really shouldn't have forgotten about, which might be the relationship that i was searching for in some of the recent instances...

so let's see, consider adjoint 1-cells in a 2-category... ??and consider the yoneda embedding of the 2-category... ???any 2-functor preserves adjoints... ??...
??maybe some variance / level confusion here??... flip/slip ... ??...

??so for example with adjoint objects in a monoidal category, tensoring with an adjoint of an object x provides an adjoint to tensoring with x, thus giving an "internal homming from x" functor ... ?? so then [x,1] = 1 tensor adjoint(x) = adjoint(x) ... ??...


?????so what were the recent instances?? one was in discussing "absolute colimits" (or something...) with mike shulman on the n-category cafe... "half-exact functor induces homotopy-exact ..." ... i should work this stuff out...

there's something suggestive here about absolute colimits and ... ??adjoint objects as a strengthened form of a certain sort of internal hom ... ???maybe really close connection here???....

in that discussion with shulman (and also that guy richard something-or-other...) i also found myself bumping into ideas that i tried to learn from todd... ??related to things like "cartesian bicategories" which as a matter of fact i'm again trying to learn from him...

also, this business about line objects and so forth... "good embedding" / "good epi" in an algebraic-geometric theory... "quasi-regular epi" ...??...