  We construct a flagged $\infty$-category ${\sf Corr}$ of $\infty$-categories
and bimodules among them. We prove that ${\sf Corr}$ classifies exponentiable
fibrations. This representability of exponentiable fibrations extends that
established by Lurie of both coCartesian fibrations and Cartesian fibrations,
as they are classified by the $\infty$-category of $\infty$-categories and its
opposite, respectively. We introduce the flagged $\infty$-subcategories ${\sf
LCorr}$ and ${\sf RCorr}$ of ${\sf Corr}$, whose morphisms are those bimodules
which are \emph{left final} and \emph{right initial}, respectively. We identify
the notions of fibrations these flagged $\infty$-subcategories classify, and
show that these $\infty$-categories carry universal left/right fibrations.
