Sheaves can, like all modern mathematical constructions and abstractions, be counterintuitive beasts but, like all such constructions, a few examples can allow one to visualise them simply as a generalisation of a natural object (ie. the sets of local functions on a topological space).
I have recently become rather interested in cosheaves- a sort of natural iteration of the sheaf concept- apply the sheaf functor once, and it's contravariant, twice (being terribly careful about covers being preserved in a sensible way) and one gets a covariant construction.
My problem is that this is rather hard to anchor to anything really natural as 'sheaves over sheaves' seems rather difficult to visualise without tying ones brain in knots. So my question, in its broadest terms is 'what do cosheaves look like?' (are they for example, just $\mathcal{F}^{op}$ to some sheaf $\mathcal{F}$ by analogy with various other 'co-constructions'), but more realistically (assuming that I am making assumptions) it is "Can one find a canonical example of a cosheaf that is 'visualisable' in some sense?"