Yoneda's lemma for internal higher categories

Abstract

We develop some basic concepts in the theory of higher categories internal to an arbitrary $\infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal categories.

Simplified the proof of the Grothendieck construction; removed a flaw in the proof of Yoneda's lemma and spelled out a few lemmas to this proof wich have been left implicit in the previous versions. Changed some terminology and removed typos. Restructured the last part of the paper