Massive stars can determine the evolution of molecular clouds by eroding and photo-evaporating their surfaces with strong UV radiation fields. Thus, probing the fundamental structure of nearby molecular clouds is crucial to understand how massive stars shape their surrounding mediums and how fast molecular clouds are destroyed. By combining data from the 12m array, the Atacama Compact Array (ACA) and Total Power, we present CO and HCO+ ALMA observations of the Horsehead edge at 0.5'' angular resolution, corresponding to physical scales of about 200 astronomical units. We find that CO and HCO+ are present at the edge of the cloud, very close to the ionization (H+/H) and dissociation fronts (H/H2), suggesting a very thin layer of neutral atomic gas and almost no CO-dark H2 gas at the molecular cloud edge. Additionally, using CO as a proxy of the C+/C/CO front, we conclude that the distances between the fronts can be reproduced by isobaric stationary models, which confirms the presence of a steep density gradient, as suggested by previous observations. Still, dynamical effects cannot be completely ruled-out, and even higher angular observations will be needed to unveil their role.