  This is the second of two papers devoted to the proof of conformal invariance
of the critical double random current on the square lattice. More precisely, we
show convergence of loop ensembles obtained by taking the cluster boundaries in
the sum of two independent critical currents (both for free and wired boundary
conditions). The strategy is first to prove convergence of the associated
height function to the continuum Gaussian free field, and then to characterize
the scaling limit of the loop ensembles as certain local sets of this Gaussian
Free Field. In this paper, we derive crossing properties of the discrete model
required to prove this characterization.
