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# An Ex functor for the contravariant homotopy structure

I'm going to slack on the background and get to the point:

Is there a good notion of an $Sd/Ex$ adjunction for $sSet/S$ equipped with the contravariant model structure (cofibrations are monomorphisms and fibrant objects are right fibrations over $S$) for an arbitrary simplicial set $S$? (Note: This is in the unmarked case.)

It seems to me that any sort of naive way of doing this (for instance, by pulling back the results in $sSet$ (that is, given an object $p:X\to S$ of $sSet/S$, let

$$Ex_S(p):=S\times_{Ex(S)} Ex(X) \to S$$

with morphisms determined by the universal property)) is doomed to fail, since it does not incorporate the asymmetry of the model structure (that is, if that worked, it would also work for the covariant model structure, which seems like it shouldn't be true).

One problem with trying to mimic the classical argument is that the classical/Quillen/Kan homotopy structure (this comprises the data of the model structure on sSet and all of its relativizations sSet/S for every simplicial set S (see Cisinski's book Les Prefaisceaux comme modeles des types d'homotopie (Ch 1.3) for a precise definition) has the property of completeness, which is essentially the property that the weak equivalences of $sSet/S$ (with the Kan/Quillen model structure) are exactly the morphisms that map to weak equivalences under the canonical projection functor sSet/S -> sSet (see Ch 2.1-2.2 of Cisinski's book or Ch.3.4 of Goerss-Jardine for a systematic development of the classical case). Since the contravariant homotopy structure does not have this property, it seems imprudent to expect to be able to pull back results from "deeper" bases naively.

The definition of the contravariant homotopy structure makes it seem like we might be able to do something by putting cones in the right place, but this is just vague intuition.