Briefly, my question is the following:

Can we recognise when a simplicial model category $\def\cM{\mathcal M}\cM$ is an absolute distributor, using only the language of (simplicial) model categories?

Here, I am using the notion of absolute distributors in the sense of Lurie's $(\infty, 2)$-categories and the Goodwillie calculus I, which I recall below.

Definition 1. A fully faithful inclusion of $\infty$-categories $\def\sX{\mathscr{X}}\def\sY{\mathscr{Y}}\sX\subseteq\sY$ is called a distributor if it satisfies the following conditions:

1. $\sX$ and $\sY$ are locally presentable,
2. $\sX$ is stable under small limits and colimits in $\sY$,
3. For any morphism $\phi:x\to y$ in $\sY$ with $x\in\sX$, the pullback functor $\phi^*:\sX_{/x}\to\sY_{/y}$ preserves small colimits,
4. Either of the following equivalent conditions hold (see Corollary 1.2.5 in Lurie's work linked above):
   • the functor $\chi:\sX\to\def\Cat{\mathbf{Cat}}\def\op{\mathrm{op}}\Cat_\infty^\op$ which acts by sending $x\mapsto\sY_{/x}$ (and sending morphisms to pullbacks) preserves small limits
   • for every natural transformation $\bar\alpha:\bar p\Rightarrow\bar q:K^\triangleright\to\sY$ such that $\bar q$ is a colimit diagram in $\sX$ and $\bar\alpha|_K$ is cartesian (meaning all naturality squares are pullback squares), then $\bar\alpha$ is cartesian iff $\bar p$ is a colimit diagram

If $\sX$ is equivalent to the $\infty$-category $\def\sS{\mathscr{S}}\sS$ of spaces, then $\sY$ is called an absolute distributor.

Definition 2. A simplicial model category $\cM$ is called an absolute distributor if:

1. $\cM$ is combinatorial and left proper (which guarantees that the underlying $\infty$-category is locally presentable),
2. its underlying $\infty$-category $N(\cM^\circ)$ is an absolute distributor in the sense of Definition 1.

To reiterate my question: can we expand part 2 of Definition 2 so that it is not written in terms of $\infty$-category theory? If this is too difficult, are there sufficient conditions on the model category $\cM$ that are not too strong---but are purely model categorical---that guarantee that $\cM$ is an absolute distributor?

Below, I include some of my own thoughts.

Definition 1, condition 2

By the adjoint functor theorem, this condition is equivalent to requiring that the inclusion $\sX\subseteq\sY$ admits both a left and a right adjoint.

First, if $\cM$ is a simplicial model category, it is necessarily tensored and cotensored over $\def\sSet{\mathbf{sSet}}\sSet$, so the (essentially unique) functor $\sS\to N(\cM^\circ)$ ought to come from a functor similar to cotensoring with the terminal object of $\cM$.

If I assume that the terminal object $1_\cM$ of $\cM$ is cofibrant, then cotensoring $(-)\odot1_\cM:\sSet\to\cM$ defines a left Quillen functor (more generally, perhaps I could take a cofibrant resolution of $1_\cM$), and perhaps I can phrase condition 2 as asserting:

Condition 2, rephrased. The functor $(-)\odot1_\cM$ defines a fully faithful right Quillen functor, where $1_\cM$ is a cofibrant resolution of the terminal object of $\cM$.

I have no trouble with assuming that the terminal object is cofibrant (or even that every object is cofibrant) in $\cM$.

Definition 1, condition 3

Suppose $\cM$ is combinatorial, left proper, and satisfies condition 2, rephrased. Allow me to abuse notation and identify $\sSet$ with its image in $\cM$ via $X\mapsto X\odot1_\cM$.

By the adjoint functor theorem again, colimit preservation in condition 3 is equivalent to asking for right adjoints. Therefore, condition 3 for the model category $\cM$ ought to say:

Condition 3, rephrased. For any morphism $\phi:M\to X$ in $\cM$, where $X$ is a simplicial set, the pullback functor $\phi^*:\sSet_{/X}\to\cM_{/M}$ is a left Quillen functor between slice model categories.

Definition 1, condition 4

This is the most puzzling axiom for me, as neither equivalent formulation seems very accessible from the model category directly. My initial thought was to define the analogous functor $\chi:\sSet\to(\sSet)\Cat^\op$ sending $X\mapsto\cM_{/X}$ and ask that it defines a right Quillen functor into the (opposite of the) model category of $\sSet$-enriched categories, but even overlooking the size issues, this functor is really a $2$-functor.

The second form of condition 4 in terms of cartesian natural transformations is likely the "better" option, but I'm not sure I know how to translate this into homotopy-natural transformations, homotopy pullbacks, and homotopy colimit diagrams.