Consider the diagram inclusions
and the induced diagram of over quasi-categories
Notice that by definition of limit in a quasi-category the quasi-categorical pullback is the terminal object in , while is the terminal object in .
The strategy now is to show that both these morphisms and are acyclic Kan fibrations. That will imply that these terminal objects coincide as objects of .
First notice that the inclusion
is a left anodyne morphism, being the composite of pushouts of left horn inclusions
We could also prove this by showing that this functor is homotopy initial using the characterization in terms of slice categories, and then invoking the theorem of HTT 220.127.116.11(4) which says (in dual form) that an inclusion of simplicial sets is homotopy initial if and only if it is left anodyne.
One of the properties of left anodyne morphisms is that restriction of over quasi-categories along left anodyne morphisms produces an acyclic Kan fibration. This shows the desired statement for .
To see that is also an acyclic fibration observe that can be factored as
Observe that fits into a pullback diagram
and hence is an acyclic Kan fibration since is one, on account of the fact that the square
is a pullback in . Finally, is a trivial fibration since
is left anodyne; clearly this is a pushout of and so it suffices to show that is left anodyne. But this map factors as and clearly is left anodyne since it is a pushout of .