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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).
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.

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 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 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).
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. $F=L_2.R_1$.

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 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).
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.

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 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).
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. $F=L_2.R_1$.

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 $L_i\dashv R_i$ yields the derived adjunction $\mathbf{L}L_i\dashv \mathbf{R}R_i$ 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 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).
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. $F=L_2.R_1$.

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 ?