[Math] Geometric Realization of a Simplicial Category

Question

Let $S:\varDelta^{op}\to (cat)$ be a functor where the category on the right is the category whose objects are categories with cofibrations and morphisms are exact functors(from Waldhausen's paper, Algebraic K-Theory of spaces). Waldhausen talks of the geometric realization of such a simplicial category. In the case of a simplicial set, I know how to construct the geometric realization. However, in this case, $S_n$ is a category. It is not clear to me if he is assuming that this is a small category for each n, in which case we could proceed to construct the geometric realization as in the case of a simplicial set. If each $S_n$ is not a small category, then is it still possible to define the geometric realization of $S$?

Answer 1

As Buehler states in the comments, Waldhausen is taking the nerve degreewise, and then taking the diagonal of the resulting bisimplicial set. This is a model for the homotopy colimit of the simplicial diagram of nerves.

Waldhausen mentions the question of smallness himself in a remark on p. 14 of Algebraic K-theory of spaces. As he observes, it is only necessary to assume that his categories with cofibrations and weak equivalences are small up to weak equivalence.