By finitely generated contramodule, I meant a finitely generated object in the category of contramodules. In this sense, can a contramodule be expressed as direct limit of its finitely generated subcontramodules? We know that a comodule over a coalgebra over a field can be expressed as direct limit of its finitely generated subcomodules (by finitely generated comodule, I mean finitely generated as a vector space which in this case is equivalent to the notion of a finitely generated object in the category of comodules). So, I am thinking if something like this holds for contramodules too.

I understand that all colimits exist in the category of contramodules and that finite colimits agree with those in vector spaces. I was hoping if there is a way to explicitly understand those colimits (say, arbitrary coproducts, as you mentioned) or if there is some functor that one applies on the colimit taken in the vector space to obtain the colimit in the contramodule category. For instance, the category of quasi-coherent sheaves have all arbitrary products and those are obtained by applying a functor called the 'coherator' to the product taken in larger category of sheaves of modules.

@Neil Epstein : Could you provide me with a reference where I can find that another version of the statement that is true, which subsumes the surjection example?

Yes, thanks! I got lost in the H- module stuff and missed this simple thing!

What I mean is when you evaluate the sequence on $V \rightarrow \mathcal{X}$, you have to deal with $V \times_{\mathcal{X}}U \rightarrow \mathcal{X}$. How does the existence of $Z$ helps? Also, if small smooth site is same as lisse-etale site, why is this difference in terminology?

@MarcHoyois: I do understand that for the smooth surjection $V\times_{\mathcal{X}} U\xrightarrow{smooth} V$ there exists some scheme $Z$ such that there is a map from $Z \rightarrow V\times_{\mathcal{X}} U$ and an etale surjection $Z \rightarrow V$ such that the diagram commutes. But I don't see how that makes the sequence mentioned in my question exact! Could you explain a bit?