The Stacks project

If we want to follow the proof of Lemma 12.4.2, I think we should consider the projection onto $G$.

Lemma 002K (lemma-functorial-colimit) is somewhat overkill here. Suggested proof (to get rid of the "omit"):

1. Wait for material on Karoubi envelopes, in particular their (2-)universal property, to be added to Section 09SF (section-karoubian) (cf. Comment 8065).

2. Observe, as a consequence of the aforementioned (2-)universal property, that if $\mathcal{C} \subset \mathcal{D}$ is fully faithful and $\mathcal{C}$ is Karoubian, then a (finite) biproduct of objects in $\mathcal{D}$ lies in $\mathcal{C}$ iff each biproductand lies in $\mathcal{C}$. (This probably also belongs in Section 09SF (section-karoubian).)

3. Observe that $\text{coCones} ( c ; F \oplus G ) \simeq \text{coCones} ( c ; F ) \oplus \text{coCones} ( c ; G )$ where $\text{coCones} ( - ; X ) : \mathcal{C} \to \textit{Ab}$ is the functor mapping $c \in \text{Ob} (\mathcal{C})$ to the Abelian group of cones under $X$ with vertex $c$.

4. Conclude by that $\mathcal{C} ^ {\text{opp}} \subset \text{Fun} ( \mathcal{C} , \textit{Ab} )$ is a Karoubian fully faithful subcategory (implicitly using that Karoubianness is a self-dual notion) that $\text{coCones} ( c ; F \oplus G )$ is "representable" iff $\text{coCones} ( c ; F )$ and $\text{coCones} ( c ; G )$ are "representable", hence that $\text{colim} ( F \oplus G )$ exists iff $\text{colim} (F)$ and $\text{colim} (G)$ exist.

("Representable" is in scare quotes because it seems that representability/Yoneda aren't treated within Stacks separately in the $\textit{Ab}$-enriched setting. Of course, the relevant results/proofs are vritually the same as in the case of $\textit{Sets}$. As an expositional question I'm not sure what the most parsimonious way to navigate this is.)