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For an object X in a category, the representable functor $y(X)$ maps any object to the set of morphisms into X, denoted as Hom(-, X). It captures all the relationships (morphisms) from other objects to X.

Conditions: X is an object in a category.

For an object X in a category, the representable functor $y(X)$ maps any object to the set of morphisms into X, denoted as Hom(-, X). It captures all the relationships (morphisms) from other objects to X.

Conditions: X is an object in a category.

The Yoneda lemma states that for any functor F from C^{op} to Set, the value $F(X)$ is isomorphic to the set of natural transformations from the representable functor Hom(-, X) to F. Informally, it means an object is entirely determined by its relationships to other objects.

Conditions: C is a category.; X is an object in C.; F : C^{op} $\to$ Set is a functor.

The Yoneda lemma states that for any functor F from C^{op} to Set, the value $F(X)$ is isomorphic to the set of natural transformations from the representable functor Hom(-, X) to F. Informally, it means an object is entirely determined by its relationships to other objects.

Conditions: C is a category.; X is an object in C.; F : C^{op} $\to$ Set is a functor.