Yoneda lemma

The Yoneda lemma gives natural bijectionsand their contravariant analogues.

For a presheaf on a category, the categorical coend identifies $(fh,x)$ with $(h,X(f)x)$. The displayed isomorphism sends the class of $(h:U\to W,x\in X(W))$ to $X(h)x$, with inverse $y\mapsto [1_U,y]$. The coend relation makes these inverse and natural. Consequently every presheaf is the colimit of representables indexed by its category of elements.

The Yoneda embedding sends $A$ to $\mathcal{C}(-,A)$. The Yoneda lemma identifies natural transformations $yA\to yB$ with morphisms $A\to B$, so the embedding is full and faithful. It preserves every limit that exists in $\mathcal{C}$.

For $f:A\to B$, the Yoneda embedding sends $f$ to postcomposition $\mathcal{C}(-,A)\to \mathcal{C}(-,B)$. If this is epic in the presheaf category, its component at $B$ is surjective, so $1_B=fg$ for some $g:B\to A$. Conversely such a section makes the natural transformation split epic. Pointwise surjectivity follows directly by doubling the target presheaf along the image subpresheaf: the two inclusions agree on the image but differ on any element outside it.

For a small Cartesian closed category, the Yoneda embedding into its presheaf category preserves exponential objects. At $C$, the exponential in a presheaf category has value $\mathrm{Nat}(YC\times YA,YB)$. Since $YC\times YA\cong Y(C\times A)$, the Yoneda lemma identifies this with $\mathcal{C}(C\times A,B)\cong \mathcal{C}(C,B^A)$. Every identification is natural in all three objects.

The covariant version of the Yoneda embedding sends an object to its outgoing representable functor. Precomposition reverses arrows. The Yoneda lemma gives $\mathrm{Nat}(Y(A),Y(B))\cong \mathcal{C}(B,A)$, so it is full and faithful. It carries existing colimits in $\mathcal{C}$ to pointwise categorical limits.