I am struggling hard to understand the pushforwards and pullbacks of cosheaves. Are they also cosheaves? And what are quasicoherent cosheaves? Is there anything like coquasicoherent cosheaves? Please tell me a good refernce on theses topics, if there is some.
edit: I was assuming you wanted an equalizer sheaf property, but this is not the definition of cosheaf, see comments - the following has nothing to do with cosheaves then!
If by "cosheaf" you mean a covariant functor from the opens of a space to sets/groups/etc., you could look at Moerdijk/MacLane's "Sheaves in Geometry and Logic" - there you can learn some general sheaf theory on sites, which includes the cosheaf case. In particular pushforward and pullback are transport along the the two functors comprising a "geometric morphism". The notion of quasicoherent (co)sheaf can maybe also be defined in this generality by saying that something should look locally like a pullback from the "base topos".
Sorry for the jargon, I didn't get from your question what exactly you are up to - just take a look at the book and see if it suits you.
I've written a preprint about (what I call) contraherent cosheaves of modules over the structure sheaf of rings of a scheme -- http://arxiv.org/abs/1209.2995 . These are a kind of dual creatures to quasi-coherent sheaves. The preprint will be updated and expanded (eventually).
If you have the stomach for hard topos theory a good reference is
Singular coverings of toposes -- M. Bunge, J. Funk
The first chapter is probably enough to answer your question. The upshot is that if you have a site, then the category of cosheaves can be identified with the category of cocontinuous functors on the category of sheaves. This should give you a pretty good idea of what operations you can perform on cosheaves.
Btw, in this context, cosheaves are also called Lawvere distributions -- distributions because of the analogy with the Riesz representation theorem that identifies measures (cosheaves) with linear functionals (cocontinuous functors).
Hope it helps, regards, G. Rodrigues
According to Skliarienko, the Borel-Moore homology (with coefficients in a sheaf) is badly "defective"; and the "right" homology should have coefficients in a co(pre)sheaf. (He makes this point in a few of his works; but most clearly in the editor's comments to the Russian translation of Bredon's "Sheaf theory".)
In fact, there exist two papers by Beniaminov which are precisely about this. I'm a bit puzzled that Skliarienko cited them with enthusiasm in early 1970s, but later ignored on several occasions when speaking of the desired cosheaf homology. There is also a followup paper by Golovin which I haven't checked yet.