Decorated marked surfaces II: Intersection numbers and dimensions of Homs

We study the 3-Calabi-Yau categories $\mathcal{D}$ arising from quivers with potential associated to a decorated marked surface ${\mathbf{S}}_{△}$ introduced by the first author. We prove two conjectures in the prequel, that under a bijection between certain objects in $\mathcal{D}$ and certain arcs in ${\mathbf{S}}_{△}$, the dimensions of morphisms between these objects equal the intersection numbers between the corresponding arcs.