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$\def\C{\mathsf{C}} \def\D{\mathsf{D}} \def\hoc{\mathsf{HoC}} \def\hod{\mathsf{HoD}} \def\L{\mathbf{L}} \def\R{\mathbf{R}}$I'm a little confused by [R, Remark 6.4.14]. Before stating the remark, I'll introduce the jargon: A homotopical category is a category equipped with a class of morphisms (named, ‘the weak equivalences’) that satisfy the 2-out-of-6 property [R, Definitions 6.4.1, 6.4.3]. Given a homotopical category $\C$, its homotopy category, $\hoc$ is defined to be the strict localization of $\C$ with respect the weak equivalences (that is, there is a functor $\C\to\hoc$ which is initial in the category of functors under $\C$ that send all weak equivalences to isomorphisms) [R, Definition 6.4.4]. Given a functor $F:\C\to\D$ between homotopical categories, if the right Kan extension of $\C\xrightarrow{F}\D\to\hod$ along $\C\to\hoc$ exists, we denote it by $\L F$ and call it the total left derived functor of $F$ (dually, one defines the total right derived functor $\R G$ of $G:\D\to \C$) [R, Definition 6.4.8].

[R, Proposition 6.4.13]. Suppose $F\dashv G$ is an adjunction between homotopical categories and suppose also that $F$ has a total left derived functor $\L F$, $G$ has a total right derived functor $\R G$, and both derived functors are absolute Kan extensions. Then the total derived functors form an adjunction $\L F \dashv \R G$ between the homotopy categories.

(Recall that a Kan extension is said to be absolute whenever it is preserved by any functor departing from the target of the extension, in the sense of [R, Definition 6.3.1].)

Here's where I have the issue:

[R, Remark 6.4.14]. If $F\dashv G$ is an adjunction satisfying the hypotheses of Proposition 6.4.13, then the adjunction diagram depicting functors \gamma: C\to HoC, \delta: D\to HoD, LF:Hoc\to HoD, RG:HoD\to HoC, LF\dashv RG between the total derived functors is the unique adjunction compatible with the localization functors $\gamma:\C\to\hoc$ and $\delta:\D\to\hod$ in the sense that the diagram $$ \label{diag}\tag{1} \require{AMScd} \begin{CD} \D(Fc,d)@>\cong >>\C(c,Gd)\\ @V\delta VV@VV\gamma V\\ \hod(Fc,d)@.\hoc(c,Gd)\\ @V Fq^*VV@VV Gr_*V\\ \hod(\L Fc,d)@>\smash\cong >>\hoc(c,\R Gd) \end{CD} $$ commutes for each pair $c\in\C$, $d\in\D$.

My only question is: why is this true? Moreover, what are $Fq$ and $Gr$ in the last diagram? The remark states we are operating under the hypotheses of Proposition 6.4.13. Nonetheless, the choice of the notation $q$ and $r$ makes it seem we need the hypotheses of [R, Proposition 6.4.11], which require the existence of a so-called left deformation $q$ and right deformation $r$ [R, Definition 6.4.10].

References

[R] E. Riehl, Category Theory in Context

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$\def\C{\mathsf{C}} \def\D{\mathsf{D}} \def\hoc{\mathsf{HoC}} \def\hod{\mathsf{HoD}} \def\L{\mathbf{L}} \def\R{\mathbf{R}}$We don't need the hypotheses of [R, Proposition 6.4.11], only those of [R, Proposition 6.4.13] suffice.

Before stating [R, Proposition 6.4.11], Riehl credits the result to [M]. It is in the statement of [M, theorem] where we find what we need to deduce commutativity of \eqref{diag}. In [M, theorem] it is said that the the (co)unit morphisms $𝛆:\L F\cdot \R G\to 1_\hod$ and $𝛈:1_\hoc\to\R G\cdot \L F$ may be picked such that the following squares commute: $$ \label{squares}\tag{$*$} \require{AMScd} \begin{CD} \L F\cdot \delta\cdot G@>\alpha G>> \gamma\cdot F\cdot G\\ @V(\L F)\beta VV@VV\gamma\varepsilon V\\ \L F\cdot\R G\cdot \gamma@>\smash{𝛆\gamma}>>\gamma \end{CD}\qquad\qquad \begin{CD} \R G\cdot \gamma\cdot F@<\beta F<< \delta\cdot G\cdot F\\ @A(\R G)\alpha AA@AA \delta\eta A\\ \R G\cdot \L F\delta@<\smash{𝛈\delta}<< \delta \end{CD} $$ where $\alpha:\L F\cdot\delta\to \gamma\cdot F$ and $\beta:\delta\cdot G\to\R G\cdot\gamma$ are the comparison transformations. But this means exactly that the data $(F,G,\L F,\R G,\alpha,\beta)$ defines a lax morphism of adjunctions [ref]. Thus, as it is explained in the linked answer, the diagram $$ \label{diag'}\tag{1'} \require{AMScd} \begin{CD} \D(Fc,d)@>\cong >>\C(c,Gd)\\ @V\delta VV@VV\gamma V\\ \hod(Fc,d)@.\hoc(c,Gd)\\ @V \alpha_c^*VV@VV \beta_{d,*}V\\ \hod(\L Fc,d)@>\smash\cong >>\hoc(c,\R Gd) \end{CD} $$ commutes. Lastly, uniqueness as asserted in [R, Remark 6.4.14] comes from the fact that if we have some adjunction $\L F\dashv \R G$ such that \eqref{diag'} commutes, then by [ref] the squares \eqref{squares} commute. But such $𝛆$, $𝛈$ are unique, as it is shown in the proof of [M, theorem].

References

[R] E. Riehl, Category Theory in Context

[M] G. Maltsiniotis. Le théorème de Quillen, d’adjonction des foncteurs dérivés, revisité. C. R. Math. Acad. Sci. Paris, 344(9):549–552, 2007

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