Checking if an arrow in a commutative diagram is a quasi-isomorphism This problem has come up in my research, but I figured I'd try here rather than mathoverflow first, since it's likely a simple homological algebra problem (my homological algebra knowledge is entirely self-taught).
I have the following commutative diagram of complexes (of sheaves), where arrows $\xrightarrow{\sim}$ denote quasi-isomorphisms, $\twoheadrightarrow$ denote epimorphisms, and $\rightarrowtail$ denote monomorphisms. The left column is just the canonical split short exact sequence for a direct sum.
In case it matters, these are complexes of sheaves on a manifold: $A_1$ and $A_2$ are locally constant sheaves, and $A_1'$, $A_2'$, and $B$ are soft resolutions by real smooth vector bundles and differential operators, and $L$ is a differential operator.


Question: How can I verify if the arrow at the bottom $A_2 \to \operatorname{Ker}(h \circ L)$ is a quasi-isomorphism? More generally, is there anything I can say about the complex $\operatorname{Ker}(h \circ L)$ here?

 A: Unfortunately, I believe there's essentially nothing you can say here -- given what you've shown so far, at least.  Here's the essential problem: it's homotopically meaningless in and of itself to say that a morphism of chain complexes is mono or epi.  Indeed, any map is equivalent to a mono up to quasi-iso, and similarly any map is equivalent to an epi up to quasi iso.  Compare this with the constructions in topological spaces which turn any map into a cofibration (the mapping cylinder construction) and a fibration (the mapping path space construction).
Rather, the utility of these special classes of morphisms lies in what they guarantee you, if and when you perform a universal construction.  For instance, If a map $X \to Y$ of topological spaces is a closed inclusion, then its cofiber (the pushout of $pt \leftarrow X \to Y$) is invariant up to weak homotopy equivalence under weak homotopy equivalence in $Ar(Top)$.  That is, given a morphism from $X \to Y$ to $X' \to Y'$ whose components are both weak equivalences and such that both of these maps are closed inclusions, then the induced map on cofibers will be a weak equivalence as well.
In the situation you describe, You are not making any reference to $ker(h \circ L)$ with regards to any universal construction.  The only thing you've said, homotopy invariantly speaking, is that $ker(h \circ L)$ admits a map to the direct sum of $A_1$ with something else, through which the inclusion of $A_1$ factors.
