Is it possible to view the nerve of a small category as being the result of a monad bar construction?
https://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html#c009959
In this blog post, Todd Trimble gets somewhat close to it, but not quite. I came up with the same idea when $X = C_0$ (and the "discrete fibration" is the identity map on $C_0$) but it isn't exactly the nerve.