  Given a locally presentable category together with a suitable functorial
cylinder object, we construct model structures which are sensitive to the
`direction' of the cylinder. We show that the Covariant and Contravariant model
structures on simplicial sets as well as the coCartesian and Cartesian model
structures on marked simplicial sets are examples of our formalism. In this
setting, notions of final and initial maps and smooth and proper maps arise
very naturally and we will identify these maps in the examples.
