Goldblatt gives a brief overview of adjunctions in his "Topoi", and one of the exercises asks to characterise the partial arrow classifier in terms of some universal arrow.
Well, I gave it some thought, and I'm not even sure where to go from there. The main problem I have is that in the "there exists" part I have only one arrow (the PAC itself), and in the "such that for every" part I'm given two arrows (the "top-left" components of the PAC pullback), and I'm just not sure how to construct a category and a functor into/from it that'd allow for both. I tried something along the lines of the diagonalization functor used for (co)products derivation via adjoints, but it didn't get me far (or anywhere, frankly).
So, how would I do that?
Again, my exposition to adjoints is limited to several pages of "Topoi", so I likely miss some otherwise well-known results.