categories fibered in small categories

[Andrew Salch <asalch@math.jhu.edu>]

I have a question for the categories list: it is well-known (and proven
e.g. in Hollander's PhD thesis) that categories fibered in groupoids over
a small category C are equivalent to lax presheaves of groupoids on C. I
would like to use the generalization of this result in which the word
"groupoids" is replaced throughout by "small categories." It is not hard
to write out how this proof goes, but I suspect there is some vast
generalization of this, e.g. with the word "groupoids" replaced by
"quasicategories" or something of that nature, which somebody has already
proven, and in that case I would prefer to cite the more general result.
Does anyone know if something like this is already in the literature?

Thanks,
Andrew S.


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