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I'm actually a little sad to see close votes. I have a sense that this is not likely to be answered at Math.SE, not because nobody there can do it, but because it will get lost in the crowd.
@BenjaminSteinberg I have a feeling auto-correct interfered more than once in your first comment. I couldn't parse "representing the end in a sheaf..."
Some years back, an incident arose where it looked like some monetary incentive was in the offing, and it didn't seem the community was much in favor. I think it's not a bad idea just asking your research question at MO without mentioning money, and see what happens.
A reference is a start, but could you please point to specific results in the reference that address the question precisely? We want you to show us your insights! (Sorry, what do 22 and 498 signify?)
Oh, I see your difficulty. You should just use Paquette's definition and add the Murfet-compactness condition to it (or the Lurie-compactness condition, depending on what you are trying to do). I think a more conceptual way of defining Murfet-compactness is that an object $M$ is compact if $\hom(M, -): C \to Ab$ preserves coproducts (we're assuming here, as I think Murfet intends, that our categories here are $Ab$-enriched).
I'm not sure what your second question is trying to ask. In $R$-Mod, $R$ is a compact generator since every $R$-module is a quotient of a free $R$-module (a coproduct of copies of $R$). What else are you looking for?
