Proposition 6.167.
Let \(T\) and \(C\) be presentable categories. Then the functor \(\pi_T\colon \int_T C \to T\) is a bicartesian fibration, which is classified by both of the following functors:
\[T \to \Cat, \qquad A \mapsto T_{/A} \otimes C,\]
and
\[T\catop \to \Cat, \qquad A \mapsto \Fun^{\lim}(T_{/A}\catop, C).\]
Proof sketch
Note that the first functor takes the form \(T \to \PrL\), hence its cocartesian unstraightening is always a bicartesian fibration whose cartesian straightening is the corresponding functor \(T\catop \to \PrR\) obtained by passing to right adjoints, which by Lurie's explicit formula for tensor products in \(\PrL\) is given by \(A \mapsto \Fun^{\lim}(T\catop_{/A},C)\). It thus suffices to prove the first claim.Since \(T\) is presentable, write it as an accessible localization of a presheaf category \(\PSh(D)\). Both constructions in the statement commute with such localizations, so it suffices to treat \(T=\PSh(D)\). In this case
\[\PSh(D)\otimes\int_{\An}C\simeq\Fun(D\catop,\int_{\An}C),\]
and the projection to \(\PSh(D)\) is computed pointwise. Its fiber over a presheaf \(A\colon D\catop\to\An\) is therefore the category of compatible \(C\)-valued families indexed contravariantly by the category of elements \(\El(A)\), namely \(\Fun(\El(A)\catop,C)\). On the other hand, \(\PSh(D)_{/A}\simeq\PSh(\El(A))\), so \[\PSh(D)_{/A}\otimes C\simeq\Fun(\El(A)\catop,C).\]
These identifications are natural in \(A\) and identify the cocartesian transport. Passing to right adjoints identifies the cartesian straightening with \(A\mapsto\Fun^{\lim}(T_{/A}\catop,C)\), as claimed.