Compositionality is a cognitive mechanism that allows humans to systematically combine known concepts in novel ways. This study demonstrates how artificial neural agents acquire and utilize compositional generalization to describe previously unseen images. Our method, termed "Composition through Decomposition", involves two sequential training steps. In the 'Decompose' step, the agents learn to decompose an image into basic concepts using a codebook acquired during interaction in a multi-target coordination game. Subsequently, in the 'Compose' step, the agents employ this codebook to describe novel images by composing basic concepts into complex phrases. Remarkably, we observe cases where generalization in the `Compose' step is achieved zero-shot, without the need for additional training.