This question is a follow-up to a previous question I asked.

If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\mathcal{D}(\mathsf{A})$ is equivalent to the ordinary derived category of $\mathsf{A},$ $\mathsf{D}(\mathsf{A}).$ This is shown in Lurie's Higher Algebra by describing the homotopy category in question explicitly.

From what I've read on these matters, this equivalence is more natural that what I've described above. I have not been able to make sense of this naturality, however. I was hopeful that there was a canonical morphism $\mathcal{D}(\mathsf{A})\to N_{\bullet}(\mathsf{D}(\mathsf{A})),$ corresponding to a morphism $h\mathcal{D}(\mathsf{A})\to\mathsf{D}(\mathsf{A})$ by adjunction which is an equivalence. But this doesn't seem to be the case, since the comparison maps that I am aware of which relate the ordinary nerve to the differential graded nerve seem to go in the wrong direction.

**Question 1:** Does such a natural/functorial comparison exist, and if so, how does one construct it?

**Question 2:** Most likely, this will follow formally from question 1, but if $\mathcal{D}_{\mathrm{qc}}(X)$ denotes the derived $\infty$-category of quasi-coherent sheaves on an ordinary scheme $X,$ then can we obtain a canonical/functorial equivalence $h\mathcal{D}_{\mathrm{qc}}(X)\simeq\mathsf{D}_{\mathrm{qc}}(X)$ as well? What about if $X$ is an algebraic stack?

**Question 3:** My last question is regarding the homotopy category of a derived algebraic stack $\mathcal{X}.$ If we view $\mathcal{X}$ as a sheaf on the 'etale site of the $\infty$-category of animated rings valued in spaces, let $\mathcal{X}_{\mathrm{cl}}$ denote the higher algebraic stack obtained by restricting to the category of discrete commutative rings inside animated rings, and let $\tau_{\leq1}\mathcal{X}_{\mathrm{cl}}$ denote the classical algebraic stack obtained from $\mathcal{X}_{\mathrm{cl}}$ by truncating every $\mathcal{X}(A)$ to obtain an ordinary groupoid. Is it known whether some statement like $h\mathcal{D}_{\mathrm{qc}}(\mathcal{X})\simeq\mathsf{D}_{\mathrm{qc}}(\tau_{\leq1}\mathcal{X}_{\mathrm{cl}})$ true in a natural way, as in the previous questions?

*Note:* I know there's not a canonical definition of a derived algebraic stack. I would be happy for a result using any definition (as long as the definition is specified), and even happier if it held for more than one definition. In question 3 I assumed that the construction $\tau_{\leq1}\mathcal{X}_{\mathrm{cl}}$ will be an algebraic stack if $\mathcal{X}$ is a derived algebraic stack, although I don't know if this is always true for any of the common definitions of derived algebraic stack. A definition of derived algebraic stack where this is always true would be ideal.

**Edit:** To be more specific about what such a comparison should satisfy, there should be an equivalence $\gamma_X : h\mathcal{D}_{\mathrm{qc}}(X)\to\mathsf{D}_{\mathrm{qc}}(X)$ (or in the other direction, if that's more natural) for every [affine] scheme/algebraic stack $X,$ such that given a morphism of schemes/stacks $f : X\to Y,$ the following diagram commutes:
$$\require{AMScd}\begin{CD}
h\mathcal{D}_{\mathrm{qc}}(Y) @>{\mathbf{L}f^*}>> h\mathcal{D}_{\mathrm{qc}}(X)\\
@VV{\gamma_Y}V @VV{\gamma_X}V\\
\mathsf{D}_{\mathrm{qc}}(Y) @>>{\mathbf{L}f^*}> \mathsf{D}_{\mathrm{qc}}(X).
\end{CD}$$