Another way to think about the Yoneda lemma is in terms of universal things. Consider, for instance, the existence of classifying spaces for bundles. The statement is that for any suitable group G, there is a space BG such that for any nice enough space X, homotopy classes of maps X → BG are in natural bijection with isomorphism classes of G-structured bundles over X. In categorical terms, that means there is a natural isomorphism between the functors

X ↦ {G-structured bundles over X}

and

X ↦ [X,BG]

The Yoneda lemma implies that this natural isomorphism is uniquely determined by a specific G-structured bundle over BG. That is, the existence of a "classifying space" BG with the above property implies the existence of a universal bundle EG → BG such that every bundle over any space X is the pullback of the universal one along a map X → BG, unique up to homotopy.

The search for representing objects, and hence for universal data, lies at the heart of a lot of modern algebraic topology, algebraic geometry, and even category theory.