$thm:taking-summands$, (2) ⇔ (3). Forwards, the same decomposition applied with $H := F$ exhibits a linear combination determined by $≡[\mathcal{F}]$ whose coefficients $hom(∐_{i ∉ t} C i, F)$ are positive; after grouping the summands by isomorphism type the lemma on determined linear combinations places every sub-union in $cl \mathcal{F}$. Backwards, (1) ⇒ (2) applied to $cl \mathcal{F}$ suffices, since $≡[\mathcal{F}]$ and $≡[cl \mathcal{F}]$ are the same relation.