Given a 2-category (i.e. bicategory) $C$ there is a unitary geometrical nerve whose 0-simplices are objects of $C$, 1-simplices are 1-arrows of $C$, 2-simplices are 2-commutative triangles (in certain orientation) of $C$.
If $C$ is a $(2,1)$-category/2-groupoid, then the nerve is a weak Kan complex (i.e. quasicategory)/Kan complex satisfying unique horn filling conditions above dimension $2$.
The converse is also true, that is we have a recognition principle. Given a weak Kan complex/Kan complex satisfying desired unique horn filling conditions, then there is a $(2,1)$-category/2-groupoid whose nerve is the given simplicial set.
This is quite amazing. But there is a subtle point here: the composition of 1-arrows is only defined up to a unique 2-arrow (or homotopy). One can choose a composition, and the choice actually does matter to much.
My question is: if we use anafuctors to replace functors, can we obtain a well-defined composition? Moreover, can we obtain an equivalence between weak Kan complexes/Kan complexes satisfying desired horn filling conditions and weak category/weak groupoid objects satisfying certain conditions in the 2-category of anafunctors?
(Anafuctors are used to avoid the axiom of choice. There are situations in which a functor is defined up to isomorphism, and we can cook up a anafunctor to avoid the choice.)