$thm:taking-summands$, (1) ⇒ (2). Writing $F$ as the disjoint union of its connected components turns $hom(F, G + H)$ into a sum, over the subsets of the components, of products $hom(∐_{i ∈ t} C i, G) · hom(∐_{i ∉ t} C i, H)$. Every graph occurring there is a summand of $F$, hence in $\mathcal{F}$, so both sides are determined by $≡[\mathcal{F}]$.