We can regard a group action on a set as a functor $$ F: BG \to Set\;, $$ where $BG$ is the category with one object and a morphism for each element of $G$, and $Set$ the category of sets.
Now, is it true that $\lim_{\leftarrow} F$ gives the fixed points of the action? If so, by duality, can we say that $\lim_{\rightarrow} F$ gives the orbit space?
The reason why I'm asking is that I've heard the statement for modules (but without proof), and I was wondering how general this is.