Let $\mathcal{A}$ be an additive category and $B \to C$ a nonzero map. Are there say "standard" techniques & criteria one should keep in mind when working with additive categories to attack the question if $B$ appears direct summand of $C$?
Note that for abelian category "the" standard technique to approach this question is the splitting lemma, unfortunately it is non appliable to additive categories as we have no kernels, cokernels, etc.
But is there a kind of "weakened replacement" of this lemma for additive categories adressing this beeing direct summand issues?

The background: The question is motivated by following particular situation appearing in context of bounded derived category of coherent sheaves. More concretely, let $f:X \to Y$ a morphism of schemes and $\mathcal{F}$ a coherent etale sheaf on $Y$ (...we work in following on etale site). Suppose that $f$ proper, and "nice enough" (eg separated &quasi-finite - so finite by properness) such that there exist a trace map

$$\text{tr}: f_{!}f^*\mathcal{F} \to \mathcal{F}$$

(reference: these notes by Brain Conrad; p54). Moreover we have canonical counit map $c:\mathcal{F} \to f_{*}f^*\mathcal{F}$ and due to assumed finiteness we have $Rf_{!}=Rf_*$ on level of derived functors, so we would obtain in the assoc derived category the composition

$$\mathcal{F} \to Rf_{!}f^*\mathcal{F} \to \mathcal{F}$$

which known to coincide with multiplication by degree of $f$, so non zero map. It looks that $\mathcal{F}$ is rather close to be included in $Rf_{!}f^*\mathcal{F}$ as direct summand.(...not sure what are the weakest assumptions on $\mathcal{F}$ making it actually true, eg constructible?) How one should argue to see it?
Here I'm also not completely sure - once having written down this composition - which feature of the derived category are we going to use to prove the splitting: Do we only need the additivity or is beeing triangulated the crucial ingrediant here? If its only a matter of additiveness, can the argumentation be "standatized" to all additive categories (as kind of "weakened version" replacing splitting lemma) or is it here something specific for those having additionally trianguated structure as it is the can for derived category? (Here we are in the situation that derived cat is only additive, so we cannot invoke the splitting lemma)