A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; which "operations" respect $E$? Note that $E$ is "extension-stable" (i.e., for any $C$-triangle $e_1\to e_2\to e_3$ the object $e_2$ belongs to $E$ if $e_1$ and $e_3$ do). Besides, $E$ is Karoubi-closed in $C$ (i.e., if $e$ is an element of $E$ then any $C$-retract of $e$ belongs to $E$ also).

I have an example of a more explicit (and general) "operation" of this sort. Let us say that $M$ is a "pseuo-extension" of an object $N$ by $A$ if there exists a distinguished triangle $A\stackrel{g}{\to} M\stackrel{f}{\to} M'$ such that $f$ factors through $N$ (this is actually equivalent to assuming the existence of a triangle $B\stackrel{g}{\to} M\stackrel{f}{\to} M'$ such that $f$ factors through $N$ and $g$ factors through $A$; so, the notion is self-dual). Then $E$ is closed with respect to "pseudo-extensions". Have anybody previously studied constructions of this sort? If I am the first to consider this notion, does "pseudo-extension" sounds fine to you?

P.S. Surprisingly, being closed under direct summands and extensions characterizes precisely those classes of objects of $C$ that are zero sets of a cohomological functor $H$ from $C$ into abelian groups (see the paper https://arxiv.org/abs/1508.04427). This probably implies that all "finite operations" of the sort described above reduce to retractions and extensions.