subcategory of ${\underline{C}}$ via $F$), and the triangle (\[ebou\]) is unique up to a canonical isomorphism.

2.  \[ibouort\] ${{}^{\perp}}{\operatorname{Obj}}{\underline{D}}={\operatorname{Obj}}{\underline{E}}$.

3.  \[ibou2\] The functor $i$ possesses an (exact) left adjoint $A$ and the morphism $g$ in (\[ebou\]) is given by the unit of this adjunction. Moreover, this unit transformation yields an equivalence of the Verdier localization of ${\underline{C}}$ by ${\underline{E}}$ (that is locally small in this case) to ${\underline{D}}$.

4.  \[ibou3\]The morphism $f$ in (\[ebou\]) is given by the counit of the adjunction $F\dashv G$, and this counit gives an equivalence ${\underline{E}}\cong {\underline{C}}/{\underline{D}}$.

IV\. A full embedding $F$ as above possesses a right adjoint if and only if for any $N\in {\operatorname{Obj}}{\underline{C}}$ there exists a distinguished triangle (\[ebou\]).

I. This is Lemma 5.3.6 of [@neebook]. II. Obvious.

III.\[ibou1\]. The existence of (\[ebou\]) is immediate from Proposition 9.1.18 (that says that the Verdier localization ${\underline{C}}\to {\underline{C}}/{\underline{E}}$ is a [*Bousfield localization*]{} functor in the sense of Definition 9.1.1 of ibid.) and Proposition 9.1.8 of ibid. (cf. also Proposition 1.5 of [@bondkaprserr]). Lastly, the essential uniqueness of (\[ebou\]) easily follows from [@bbd Proposition 1.1.9] (cf. also Proposition \[pbw\](\[icompl\]) below).

\[ibouort\]. According to the aforementioned Proposition 9.1.18 of [@neebook], we can deduce the assertion from 