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Working with a small category $\mathcal{C}$ (or in a fixed ambient universe), for a categorical presheaf $X$ and fixed $U$, apply the categorical coend construction to $T(A,B)=\mathcal{C}(U,B)\times X(A)$. It is the quotient of the disjoint union of pairs $(h:U\to W,x\in X(W))$ by the relationsDefine ${\Phi }_U([h,x])=X(h)x$. It respects every relation because $X(fh)x=X(h)X(f)x$. Define ${\Psi }_U(y)=[1_U,y]$. Then ${\Phi }_U{\Psi }_U(y)=y$. Conversely the relation with $f=h:U\to W$ gives $[h,x]=[1_U,X(h)x]$, so ${\Psi }_U{\Phi }_U$ is the identity morphism. ThereforeFor $k:V\to U$, restriction on the categorical coend sends $[h,x]$ to $[hk,x]$, and ${\Phi }_V([hk,x])=X(k){\Phi }_U([h,x])$. A natural transformation $X\to Y$ likewise acts on the second coordinate and commutes with $\Phi$. Thus the density formula for presheaves is a natural isomorphism, not just a family of unrelated bijections.

For fixed $U$, the categorical coend on the right is the quotient of ${\coprod }_W\mathcal{C}(U,W)\times X(W)$ by the identificationsDefine ${\Phi }_U([h,x])=X(h)x$. This is well defined because the contravariant functor law gives $X(fh)x=X(h)X(f)x$. Its proposed inverse sends $y\in X(U)$ to $[1_U,y]$. The composite to $X(U)$ is the identity. In the other direction the coend relation, applied with $f=h$ and initial map $1_U$, gives $[h,x]=[1_U,X(h)x]$. Thus the inverse is genuine andFor $a:U^{\prime }\to U$, the coend map replaces $h$ by $ha$; its image is $X(ha)x=X(a)X(h)x$, proving naturality in $U$. A morphism of presheaves acts on $x$ and commutes with every $X(h)$ by naturality, proving naturality in $X$. This is the density formula for presheaves.