ct.category theory - Is there a notion of "good" distributor/profunctor for model categories?

There's a natural generalization of Quillen adjunction.

Let $A$ and $B$ be model categories, and $M:A^{op}times Bto Set$ be a distributor/profunctor/module. This should be Quillen if whenever $i:ato a'$ is a cofibration in $A$ and $p:bto b'$ is a fibration in $B$, with either $i$ or $p$ a weak equivalence, then the induced map from $M(a',b)$ to the pullback $M(a,b)times_{M(a,b')}M(a',b')$ is surjective.