Compassion focuses on the Dukkha and wishes to alleviate the Dukkha. The Lovingkindness side focuses on happiness to replace the unhappiness. So that's how they work. Indeed we want to use them as a unity, we would rather not separate the two but as to how they work Compassion looks at the Dukkha, Lovingkindness looks what's going to happen after the Dukkha is gone and happiness comes instead.