Is there a proof that uses (co)ends solely to establish the derived adjoint correspondence of e.g. deformable functors? In Riehl's book "Categorical homotopy theory" (the pdf may be downloaded on https://emilyriehl.github.io/books/) Exercise 2.2.15 on page 21 is given as follows:

Suppose  $F \dashv G$ is an adjunction between homotopical categories and
suppose also that $F$ has a total left derived functor $\mathbf{L}F$, $G$ has a total right derived functor $\mathbf{R}G$, and both derived functors are absolute Kan extensions. Show that $\mathbf{L}F \dashv \mathbf{R}G$. That
is, show the total derived functors form an adjunction between the homotopy categories,
regardless of how these functors may have been constructed.

I know that the standard (formal proof) which does not make use of the precise construction of the total derived functors is the one from Georges Maltsiniotis (to be found in https://arxiv.org/abs/math/0611952). However, I wonder if there is also a different proof which makes use of the left and right Kan extension formulas expressed as co/ends. More specifically, let $F \colon \mathcal{C} \to \mathcal{D}$ and $G \colon \mathcal{D}\to \mathcal{C}$ with $\mathcal{C}, \mathcal{D}$ homotopical categories. Denote by $\mathcal{C}_\sim$ and $\mathcal{D}_\sim$ the localizations at the corresponding weak equivalences and write $\gamma_\mathcal{C} \colon \mathcal{C} \to \mathcal{C}_\sim$ and $\gamma_\mathcal{D} \colon \mathcal{D} \to \mathcal{D}_\sim$ for the associated localization functors. Since the total left and right derived functors are assumed to be absolute Kan extensions we may express them by the formulas
\begin{align*}
\mathbf{LF} = \text{Ran}_{\gamma_\mathcal{C}}(\gamma_\mathcal{D}F) \cong \int\limits_{c \in \mathcal{C}} \mathcal{C}_\sim(-,\gamma_\mathcal{C}c) \pitchfork\gamma_\mathcal{D}Fc\;, \qquad \mathbf{R}G = \text{Lan}_{\gamma_\mathcal{D}}(\gamma_\mathcal{C} G) \cong \int\limits^{d \in \mathcal{D}} \mathcal{D}_\sim(\gamma_\mathcal{D}d,-) \otimes\gamma_\mathcal{C}Gd
\end{align*}
The intuition would be (at least mine) that these formulas alone should be enough to prove the adjoint correspondence. However, I have not been able thus far to manipulate the corresponding hom-sets so as to obtain isomorphisms
\begin{align*}
\mathcal{D}_\sim(\mathbf{L}F(\gamma_\mathcal{C}c'), \gamma_\mathcal{D}d') \cong \mathcal{C}_\sim(\gamma_\mathcal{C}c', \mathbf{R}U(\gamma_\mathcal{D}d')
\end{align*}
It seems like there should be something, but it continues eluding me. So to summarize, my question is this:

Is there a proof that uses the (co)end formulas solely? Was this what Riehl intended with Exercise 2.2.15, or was she referring to the proof as given in the paper from Maltsiniotis? If there is no (known) proof that uses the (co)end machinery, I would like to know why such an approach fails, as it should encode all the relevant information of the given Kan extensions.

 A: Since you insist:
\begin{align*}
\mathcal{D}_\sim \left( \mathbf{L} F \gamma c', \gamma d' \right) 
& \cong \mathcal{D}_\sim \left( \int_{c : \mathcal{C}} \mathcal{C}_\sim \left(\gamma c', \gamma c \right) \pitchfork \gamma F c , \gamma d' \right) \\
& \cong \int^{c : \mathcal{C}} \mathcal{C}_\sim \left(\gamma c', \gamma c \right) \times \mathcal{D}_\sim \left( \gamma F c, \gamma d' \right) \\
& \cong \int^{c : \mathcal{C}} \int^{d : \mathcal{D}} \mathcal{C}_\sim \left(\gamma c', \gamma c \right) \times \mathcal{D} (F c, d) \times \mathcal{D}_\sim \left( \gamma d, \gamma d' \right) \\
& \cong \int^{c : \mathcal{C}} \int^{d : \mathcal{D}} \mathcal{C}_\sim \left(\gamma c', \gamma c \right) \times \mathcal{C} (c, G d) \times \mathcal{D}_\sim \left( \gamma d, \gamma d' \right) \\
& \cong \int^{d : \mathcal{D}} \mathcal{C}_\sim \left( \gamma c', \gamma G d \right) \times \mathcal{D}_\sim \left( \gamma d, \gamma d' \right) \\
& \cong \mathcal{C}_\sim \left( \gamma c' , \int^{d : \mathcal{D}} \mathcal{D}_\sim \left( \gamma d, \gamma d' \right) \odot \gamma G d \right) \\
& \cong \mathcal{C}_\sim \left( \gamma c', \mathbf{R} G \gamma d' \right)
\end{align*}
But I have to say that the expressions $\int_{c : \mathcal{C}} \mathcal{C}_\sim \left( \gamma c', \gamma c \right) \pitchfork \gamma F c$ and $\int^{d : \mathcal{D}} \mathcal{D}_\sim \left( \gamma d, \gamma d' \right) \odot \gamma G d$ are nonsense if interpreted literally because $\mathcal{C}_\sim$ and $\mathcal{D}_\sim$ may not have the (co)powers appearing in the formulae.
It would be better to understand the entire expressions as abbreviations for certain weighted (co)limits of $\gamma F$ or $\gamma G$ that happen to exist.
