When I previously posted about a model structure on **Cat**, I called the usual (categorical equivalence, injective-on-objects, isofibration) model structure the “standard model structure”. The *n*Lab calls it the canonical model structure on **Cat**, but I dislike that name because it always seems to suggest that there is some mechanical procedure for constructing it. As it turns out – there is!

Recall that Rezk’s **classifying diagram** for a (small) category 𝒞 is the bisimplicial set (= simplicial simplicial set) defined by **N**(𝒞)_{n, m} = Fun([*n*] × **I**[*m*], 𝒞), where [*n*] is the standard *n*-simplex in **Cat** and **I** is the groupoid completion functor (= left adjoint of the inclusion **Grpd** ↪ **Cat**). Rezk [2001] has shown that **N** : **Cat** → **ssSet** is fully faithful and is homotopically conservative in the sense of sending categorical equivalences to degreewise weak homotopy equivalences and reflecting degreewise weak homotopy equivalences as categorical equivalences. As is usual with presheaf categories, **N** has a left adjoint, namely the colimit-preserving functor *τ*_{1} : **ssSet** → **Cat** that sends the bisimplicial set representing [*n*, *m*] to the category [*n*] × **I**[*m*].

I claim that the standard model structure on **Cat** is the model structure obtained by transferring the projective model structure on **ssSet** (i.e. degreewise weak homotopy equivalences and degreewise Kan fibrations) along the left adjoint *τ*_{1} : **ssSet** → **Cat**. We have already remarked that **N** preserves and reflects weak equivalences, so it is enough to show that **N** also preserves and reflects fibrations. This is fairly straightforward.