The internal Yoneda lemma for locally Cartesian closed $\infty$-categories

Abstract

We formulate and prove internal versions of the Yoneda lemma and of the Yoneda embedding theorem in a finitely complete, locally Cartesian closed $\infty$-category $\mathscr{C}$: for every object $X\in \mathscr{C}$ and every universe $\mathscr{U}$ classifying the diagonal of $X$, the Yoneda map $\mathscr{Y}_X\colon X \to \mathscr{U}^X$ is a monomorphism. The proof uses only finite limits, dependent products and universes, and does not rely on the external Yoneda lemma. The result applies notably to every elementary $\infty$-topos, where it recovers a theorem of Rasekh [Ras18].