Let $Q$ be the homotopical category of combinatorial model categories and left Quillen functors, with left Quillen equivalences for weak equivalences.
Let $\mathbf Q$ be the corresponding $\infty$-category (in whatever foundations you prefer).
Let $Pres^L$ be the $\infty$-category of presentable $\infty$-categories and left adjoint functors.
The usual functor from relative categories to $\infty$-categories (modeled however you prefer) descends to a functor $N: \mathbf Q \to Pres^L$.
Variation A: I'm happy to work with simplicial combinatorial model categories rather than ordinary ones.
Variation B: It would also be interesting to know the answer to the following questions for combinatorial model $\infty$-categories and left $\infty$-Quillen functors. This is probably easier because one has more flexibility in this setting.
Question 1: Is $N: \mathbf Q \to Pres^L$ an equivalence of $\infty$-categories?
This functor is known to be essentially surjective -- for simplicial model categories this is in HTT, I think.
Question 2: Can the homotopical category $Q$ be refined to a model category?
There are size issues here; I'm happy with any way of handling them. I might suspect that something like Dugger's universal homotopy theories provide cofibrant resolutions. More specifically, I'd like to know:
Question 3: Are there model categories $C, D$ such that every left adjoint functor $NC \to ND$ is modeled by a left Quillen functor?