Comprehension factorization for locally small categories

Question

It is well-known that there is an orthogonal factorization system on Cat (the category of small categories and functors) where the final functors are left-orthogonal to the discrete fibrations. This factorization is called "comprehensive" in the original paper by Street and Walters.

However, in that paper the authors do not address any size issue. Can this factorization system be extended to the category of locally small categories?

Answer 1

Discrete fibrations are faithful functors, so any discrete fibration to a locally small category is locally small.

In particular if you take any comprehension factorization: $$ C \rightarrow D \rightarrow T $$

If $T$ is locally small, then $D$ is also locally small. so the answer to your question is yes.

Though you have to be aware that the kind of discrete fibrations we are talking about here can have large fibers.