Solar inverter has priority for solar charging similar to PCU. However during solar day, battery will be charged through solar but in presence of mains the load will be fed through mains through bypass mode.Solar energy stored in battery can be utilized when AC mains is absent (in Inverter mode).This way solar energy will be stored in the battery and can be utilized when AC mains gets cutoff. As long as the mains is present solar power is kept stored.
In case of PCU, solar charging of battery takes priority during solar day. Battery charges through solar until it reaches full charge voltage (typically 14.2V). After battery is fully charged, the load is fed via battery in inverter mode thus AC power is not used in presence of solar power. After the battery gets fully discharged (upto say 11V), the mains takes over-charging as well as the load is fed through mains only. This way solar energy will be used first on priority and mains energy will be utilized only after the solar energy is fully exhausted.