  For each pair of simplicial sets $A$ and $B$, the category
$\mathbf{Cyl}(A,B)$ of cylinders (also called correspondences) from $A$ to $B$
admits a model structure induced from Joyal's model structure for
quasi-categories. In this paper, we prove Joyal's conjecture that a cylinder $X
\in \mathbf{Cyl}(A,B)$ is fibrant if and only if the canonical morphism $X
\longrightarrow A \star B$ is an inner fibration, and that a morphism between
fibrant cylinders in $\mathbf{Cyl}(A,B)$ is a fibration if and only if it is an
inner fibration. We use this result to give a new proof of a characterisation
of covariant equivalences due to Lurie, which avoids the use of the
straightening theorem. In an appendix, we introduce a new family of model
structures on the slice categories $\mathbf{sSet}/B$, whose fibrant objects are
the inner fibrations with codomain $B$, which we use to prove some new results
about inner anodyne extensions and inner fibrations.
