Yoneda lemma

given a functor $F: \mathbf{Set} \to \mathbf{Set}$ and a set $\mathbf{S}$

there is an isomorphism $F(S) \cong \mathbf{Set}^{\mathbf{Set}} (y^{S}, F)$

• RHS: homset of natural transformation $y^{S} \to F$
• RHS: $y^{S}$ is the representable functor $S$
• moreover, the isomorphism is natural in both $S$ and $F$

it characters maps out of representables

• for an arbitray functor $F: \mathbf{Set} \to \mathbf{Set}$, natural transformationf $y^{S} \to F$ are in natural correspondence with elements of $F(S)$

even in a totally arbitrary cartesian category whose objects are not sets of any kind, we can still reason about them as if they were

• at lease when it comes to pairing elements and applying functions