Lemma Let be a left fibration. Then is a left fibration.
Proof: Observe that the natural functor has a right adjoint . This right adjoint is the functor which sends .
Let us say that a map in is a left fibration if the underlying map in is a left fibration. Likewise we will say that a map in is left anodyne if the underlying map in is left anodyne. Observe that a map in is a left fibration iff it has the LLP with respect to all left anodyne maps in . Therefore our task is to prove that sends left fibrations to left fibrations in , or equivalently that sends left anodyne maps in to left anodyne maps. Hence the result follows from the following result of Joyal.
Lemma (Joyal) The functor
preserves left anodyne maps.
Here is another way to think about the first lemma above. We want to prove that the map has the RLP against all horn inclusions where and . Suppose given a commutative diagram
There are two things that can happen: either , or is the cone point of . In the first case, the image of lies entirely inside , and the desired lift can be constructed since is a left fibration. In the second case, there is a unique way to extend the map to a map , i.e.\ a map . This map is a diagonal filler for the diagram above.