Are morphisms in a stable $\infty$-category generated by split injections?

Question

I've seen it stated in the $\infty$-categorical literature (without proof or reference) that every object in the $\infty$-category $\operatorname{Fun}(\Delta^1, \mathcal{C})$ of morphisms in a stable $\infty$-category $\mathcal{C}$ is a geometric realization of arrows of the form $A \rightarrow A \oplus B$. I may be missing some assumptions, but those should be "mild". Why is this the case?

Answer 1

Any map $f:A\to B$ fits in a cofiber sequence of arrows $(0\to A)\to (A\to A\oplus B)\to (A\to B)$

In other words, any map is a cofiber of split inclusions. But now cofibers (as any colimit, by the Bousfield-Kan formula) can be rewritten as geometric realizations of coproducts of the involved terms, and thus we can conclude (coproducts of split inclusions are clearly split inclusions).