Where I can find a proof of the following statement: 
,,We have two categories: $C$ and $C'$. If they are equivalent, then geometric realizations of its nerves are homotopy equivalent"? 



Homotopy equivalence of nerves
Geometric realization is a functor, so a functor $C \to C^\prime$ induces a morphism of realizations. Since a natural transformation is a functor $C \times (\bullet \to \bullet) \to C^\prime$, any natural transformation induces a homotopy. So you only need an adjunction for an equivalence of realizations. More generally, there are [Quillen's theorems A and B](https://en.wikipedia.org/wiki/Quillen%27s_theorems_A_and_B) which give an comparison of realizations based on the properties of a single functor.


