I have the following situation:

1. Three combinatorial model categories $\mathcal{K}_1, \mathcal{K}_2, \mathcal{K}_3$ such that all objects are fibrant (so no need to bother with the fibrant replacements and also they are right proper).
2. Two Quillen adjunctions $L_1\dashv R_1$ and $L_2\dashv R_2$ with $\mathcal{K}_1 \stackrel{L_1}\longleftarrow \mathcal{K}_2 \stackrel{L_2}\longrightarrow \mathcal{K}_3$
3. $L_2\dashv R_2$ is a Quillen equivalence. Moreover $R_2$ is also a (categorical) left adjoint.
4. There is a functor $F:\mathcal{K}_1\to \mathcal{K}_3$ which is neither a left nor a right adjoint; the functor $F.(-)^{cof}$ induces an equivalence of categories from $\mathrm{Ho}(\mathcal{K}_1)$ to $\mathrm{Ho}(\mathcal{K}_3)$
5. $F=L_2.R_1$.
6. $R_1$ and $R_2$ reflect weak equivalences.
7. $\mathcal{K}_3$ is left proper.
8. The three model categories $\mathcal{K}_1$, $\mathcal{K}_2$ and $\mathcal{K}_3$ are simplicial and tractable.

Does it suffice to conclude that the Quillen adjunction $L_1\dashv R_1$ is a Quillen equivalence ?

The answer is likely to be negative unless I am missing a stupid point. Indeed, the Quillen adjunctions $L_i\dashv R_i$ yield the derived adjunctions $\mathbf{L}L_i\dashv \mathbf{R}R_i$ between the homotopy categories. And the hypothesis (5) does not imply anything about the composite $\mathbf{L}L_2.\mathbf{R}R_1$, unless $R_1$ takes cofibrant objects to cofibrant objects which is plausible in my situation but extremely difficult to prove directly.

What kind of sufficient conditions could lead to this result, given the fact that the functor $L_1$ is also extremely complicated to understand ?

• I'm just between classes, with no time for a real answer, but an observation: the Quillen adjunction $(L_1,R_1)$ is a Quillen equivalence if and only if, on the level of homotopy categories, it induces an equivalence of categories. The hypothesis in (4) says that those homotopy categories are indeed equivalent (because $(L_2,R_2)$ also induces an equivalence of homotopy categories by (3)), so it's at least plausible. – David White Sep 11 '18 at 16:40
• @DavidWhite Unfortunately, I think that I cannot avoid to understand $L_1$ for which I don't know any explicit form. I just know that $L_1$ exists thanks to the theory of locally presentable categories. I wonder whether there is a characterization of a Quillen equivalence $L\dashv R$ which only uses $L$ or which only uses $R$; I am not aware of such a statement. All characterizations I know use both $L$ and $R$. – Philippe Gaucher Sep 12 '18 at 13:29