Definition. A displayed category is a lax functor $\mathcal{C} \to \mathbb{S}\mathsf{pan}(\mathsf{Set})$.
The following theorem is a generalization of the equivalence given by the Grothendieck construction between indexed categories and fibred categories. It can be found as Proposition 4 in Pavlovic-Abramsky, where they say they could not find the proof in any published work.
Theorem. $\mathcal{C}/C \cong [C, \mathsf{Span}]_{lax}$.
D. Pavlovic? and S. Abramsky?, Specifying interaction categories?, In: Moggi E., Rosolini G. (eds) Category Theory and Computer Science. CTCS 1997. Lecture Notes in Computer Science, vol 1290. Springer, Berlin, Heidelberg, 1997. DOI
Jean B Μenabou. Distributors at work, 2000. Lecture notes taken by Thomas Streicher. PDF
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