Locally finitely presentable categories have generating sets by definition. I wonder if there are any examples (or if there is a known classification) of l.f.p categories which have a cogenerating set.
Sets do, for example, and I think Boolean algebras too.
Slightly generalising the vector spaces example: If you take a module category R-Mod, over an artin algebra R (so, for example, a finite dimensional algebra), then this is l.f.p. However, it has a (finitely presented) injective cogenerator.