Image-text representation learning forms a cornerstone in vision-language models, where pairs of images and textual descriptions are contrastively aligned in a shared embedding space. Since visual and textual concepts are naturally hierarchical, recent work has shown that hyperbolic space can serve as a high-potential manifold to learn vision-language representation with strong downstream performance. In this work, for the first time we show how to fully leverage the innate hierarchical nature of hyperbolic embeddings by looking beyond individual image-text pairs. We propose Compositional Entailment Learning for hyperbolic vision-language models. The idea is that an image is not only described by a sentence but is itself a composition of multiple object boxes, each with their own textual description. Such information can be obtained freely by extracting nouns from sentences and using openly available localized grounding models. We show how to hierarchically organize images, image boxes, and their textual descriptions through contrastive and entailment-based objectives. Empirical evaluation on a hyperbolic vision-language model trained with millions of image-text pairs shows that the proposed compositional learning approach outperforms conventional Euclidean CLIP learning, as well as recent hyperbolic alternatives, with better zero-shot and retrieval generalization and clearly stronger hierarchical performance.