What are nice properties enjoyed by the coslice category $F/[P^\text{op},{\sf Set}]$ of a presheaf topos? In the example I have in mind, the object $F$ is a constant functor, say at a set $A$, but the question can more generally be phrased as "what good properties of a topos, if any, remain true for its coslices at varous objects?"

One suggestion:

A presheaf topos is locally finitely presentable (you can find a proof here). The coslice category is then again locally finitely presentable, see here.

Thanks, it is indeed true. Regularity is also preserved under coslicing (what about more general comma constructions?), so one can derive that the coslice of a topos is a regular, locally presentable category. What's the name for regular lfp categories again? (I am absolutely newbie when it comes to this kind of zoology...)

I don't know any special terminology for regular lfp categories.

The category of pointed sets is maybe the easiest example of the kind of coslice category that you mention. You can show that the initial object in this category (the one-point pointed set) is not strict. It then follows that the category of pointed sets is not extensive or distributive. So the idea of coproducts as "disjoint unions" fails a bit. Maybe some weakened form of extensiveness still holds, though.

It might be informative to consider the adjunction featuring the forgetful functor, since it behaves dually to a local homeomorphism. The forgetful functor has a left adjoint sending . Also, you can check that the forgetful functor creates all coequalizers, whence it is monadic.

So , while "almost never" a topos or anything close, is equivalent to the category of algebras of a monad on , and there is lots of theory for handling such categories.