The Stacks project

I think that maybe we can give a little bit more detail in "thus it is clear that (1) holds": Consider the commutative diagram: $$\require{AMScd} \begin{CD} X/\text{Qis}^+(\mathcal{A})@>>> X/\text{Qis}(\mathcal{A})\\ @V{F}VV@VV{F}V\\ \mathcal{D}^+(\mathcal{B})@>>>\mathcal{D}(\mathcal{B}) \end{CD}$$ Suppose the left map is essentially constant. Then we still have essentially constancy after postcomposition with the bottom map, Lemma 4.22.8. Hence, since the top map is cofinal, the right map is essentially constant. Conversely, suppose that the right map of the diagram is essentially constant. This means that there is a cocone under $X/\text{Qis}(\mathcal{A})\to\mathcal{D}(\mathcal{B})$ with vertex $Y\in\mathcal{D}(\mathcal{B})$, an object $X\to X'$ of $X/\text{Qis}(\mathcal{A})$ and a map $Y\to F(X')$ in $\mathcal{D}(\mathcal{B})$ satisfying Categories, Definition 4.22.1, (1). Since there is a quasi-isomorphism $X'\to X''$ with $X''\in K^+(\mathcal{A})$ (Lemma 13.11.5), by https://stacks.math.columbia.edu/tag/05PT#comment-8394 we can substitute $X'$ by $X''$ and assume $X'\in K^+(\mathcal{A})$. We claim that $Y\in\mathcal{D}^+(\mathcal{B})$. Since $Y\to F(X')$ is a (split) monomorphism, we have $Y\oplus Z\cong F(X')$ in $\mathcal{D}(\mathcal{B})$, by the proof of Lemma 13.4.12. Hence, since $\mathcal{D}^+(\mathcal{B})$ is strictly full and saturated in $\mathcal{D}(\mathcal{B})$ (13.6.4), we get $Y\in\mathcal{D}^+(\mathcal{B})$. Lastly, condition (2) of Categories Definition 4.22.1 for the left functor of last diagram follows from the fact that the top functor is cofinal.

Well, personally, I would have used Lemma 4.22.11 (and its dual) if I was pressed to explain more. Let's see if we get other comments on this.