I'm reading Bridgeland's *Stability conditions on K3 surfaces*. In **Lemma 4.4** there appears a full quasi-abelian subcategory $\mathscr{A} \subset \mathscr{D}$ of a triangulated category $\mathscr{D} = \mathscr{D}(X)$ of a smooth variety $X$. Then he considers a "strict" short exact sequence
$$0 \to A \to B \to C \to 0, \tag{$*$}$$
with $A, B, C \in \mathscr{A}$.

**1. Question:** How exactly does one define (short) exact sequences in a quasi-abelian category?

The definition of abelian categories $\operatorname{Ker}(f_i) = \operatorname{Im}(f_{i+1})$ seems problematic, because I don't know how to define the image. In general it seems that $\operatorname{coker ker} f \neq\operatorname{ker coker} f$, see Wikipedia.

**2. Question:** What does "strict" mean in that context? Does it have anything to do with the first question, or does it just mean that $A \neq 0 \neq B$, i.e. those are non-trivial subobjects / quotients?

After that Bridgeland goes on to argue that $f(B) = f(A) + f(C)$, where $f: K(\mathscr{D}) \to \mathbb{R}$ is an additive function.

**3. Question:** Why does a short exact sequence $(*)$ in $\mathscr{A}$ induce a triangle in $\mathscr{D}$?

I know that this is true if $\mathscr{A}$ is abelian, and $\mathscr{D} = \mathscr{D}(\mathscr{A})$ is the derived category of $\mathscr{A}$, but I see no reason why this is true in our case.