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: I'm interested in following 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 coincides 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. How one should argue to see it? Can the argumentation be "standatized" to all additive categories (as kind of "weakened version" of 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)