Proposition [efr-ZPPI]

Let \mathcal {C} be any 1-category, and let X: \Delta ^\mathrm {op} \to \mathcal {C} be a Segal object. Then there is an internal category with morphism object X[1], objects object X[0], domain and codomain given by d_1,d_0: X[1] \to X[0], identity map given by s_0: X[0] \to X[1], and composition given by X[1] \times _{X[0]} X[1] \cong X[2] \xrightarrow {d_1} X[1], where the isomorphism is the one induced by the Segal property.

This construction induces an equivalence of categories between the category \mathsf {Cat}(\mathcal {C}) of internal categories and internal functors, and [\Delta ^\mathrm {op},\mathcal {C}]_\mathrm {Segal} of Segal objects and natural transformations.