Let $\mathord{\mathbf{Set}}$ be the category of all sets with functions as the morphisms, and let $\mathord{\mathbf{Cat}}$ be the category of all small categories with functors as the morphisms.

The nerve of a (small) category $C$ is the simplicial set $\hom(i(-),C)$, where $i\colon\Delta\to\mathord{\mathbf{Cat}}$ is the fully faithful functor that takes each ordered set $[n]$ in the simplicial category, $\Delta$, to the pre-order $\mathord{\mathbf{n+1}}$. The nerve is a functor $\mathord{\mathbf{Cat}}\to\mathord{\mathbf{Set}}^{{\Delta^{\mathrm{op}}}}$.