This post is an expansion of all the details I did not have a good feeling for when I started with for page 7 of Goerss-Jardine, where the geometric realization of simplicial sets is introduced.
The construction works by constructing a few helpful categories, and using their properties along the way. Especially after unpacking the categorical results G-J rely on, there are quite a few categories floating around. I shall try to be very explicit about which category is which, and how they work.
As we recall, simplicial sets are contravariant functors from the category of ordinal numbers to the category of sets. We introduce the simplex category of a simplicial set with objects (simplices) given by maps and a map from to being given by a map in such that .