Theorem [efr-2Q3G]

Let \mathcal {C} be any 2-category . Then the coherent nerve defines a functor N : \mathsf {PsCat}(\mathcal {C})_\mathrm {ps} \to [\Delta ^\mathrm {op}, \mathcal {C}]. Moreover,

1. The coherent nerve carries fibrant pseudocategories to Reedy fibrant Segal objects.
2. A pseudofunctor of pseudocategories is an equivalence of pseudocategories if and only if it is carried to an equivalence by N.
3. Given two pseudocategories X,Y, the nerve induces a map \mathsf {PsCat}(X,Y) \to [\Delta ^\mathrm {op}, \mathcal {C}] (N(X),N(Y)) on the hom-objects in \mathcal {C}. For any fibrant X,Y, this is an equivalence. Moreover, there is a canonical strict retract. (Do we need fibrant here?)
4. Up to weak equivalence, every Reedy fibrant Segal object is the nerve of a fibrant pseudocategory, whose underlying reflexive graph is given by the restriction of

In particular, the 2-functor N: \mathsf {PsCat}(\mathcal {C})_{\mathrm {ps},\fib } \to [\Delta ^\mathrm {op},\mathcal {C}]_\mathrm {Segal}^\circ is a biequivalence.