The comprehension construction

In this paper we construct an analogue of Lurie’s “unstraightening” construction that we refer to as the “comprehension construction”. Its input is a cocartesian fibration $p\text{\hspace{0.17em}⁣}:E\to B$ between $\mathrm{\infty }$-categories together with a third $\mathrm{\infty }$-category $A$. The comprehension construction then defines a map from the quasi-category of functors from $A$ to $B$ to the large quasi-category of cocartesian fibrations over $A$ that acts on $f \colon A \to B$byformingthepullbackof$p$along$f$. To illustrate the versatility of this construction, we define the covariant and contravariant Yoneda embeddings as special cases of the comprehension functor. We then prove that the hom-wise action of the comprehension functor coincides with an “external action” of the hom-spaces of $B$ on the fibres of $p$ and use this to prove that the Yoneda embedding is fully faithful, providing an explicit equivalence between a quasi-category and the homotopy coherent nerve of a Kan-complex enriched category.