$prop:lexprod-indsub$, (1) ⇒ (2). In the formula for $hom(F, G ⋅ H)$ the factor depending on $H$ counts homomorphisms out of a disjoint union of subgraphs of $F$ induced by the classes of a partition. Each of those induced subgraphs lies in $\mathcal{F}$, so none of them distinguishes $H$ from $H'$, and the counts agree summand by summand.