Dear Andr'e, I think we can't do better than exhibiting Hom(L,-) as the directed colimit of the Set(X,-) indexed by f : L -> X.
I have a question regarding certain finite limit preserving functors Set-->Set.
If L is a locale, then the functor Hom(L,-):Set-->Set preserves finite limits, where Hom(L,X) denotes the set of morphisms of locales L-->X for a discrete locale X. Is there is a simple characterization of these flp functors?
I think I should reveil the background of my question. Look at p.7 of my https://www2.mathematik.tu-darmstadt.de/~streicher/FIBR/UniGround.pdf for formulations of my conditions Tr1-Tr3. I think Hom(L,-) validates (Tr1) and (Tr2) but presumably not (Tr3) (for EE = SS =Set). I am looking for a functor F : Set->Set validating all 3 conditions but not being equivalent to Id_Set. Butmaybe there is none? Best, Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]