Title: The comprehension construction,
Abstract: 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 $p \colon E \to B$ between $\infty$-categories together
with a third $\infty$-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 $f \colon A \to
B$ by forming the pullback of $p$ 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.