Yoneda Lemma in Double Categories

The article explains how to formulate the Yoneda lemma within double categories by defining presheaves as horizontal arrows and using profunctors instead of hom-sets. It shows that representable profunctors arise from companions and conjoints, and that the Yoneda embedding can be expressed as a vertical arrow whose action is given by a universal construction involving left Kan extensions and cartesian 2-cells. The construction ensures the embedding is dense, full, and faithful without referencing hom-sets.