= Solution
For fixed $U$, the <categorical coend> on the right is the quotient of $\coprod_W\mathcal C(U,W)\times X(W)$ by the identifications
$$
(fh,x)_V\sim(h,X(f)x)_W\qquad(h:U\to W,\ f:W\to V,\ x\in X(V)).
$$
Define $\Phi_U([h,x])=X(h)x$. 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 and
$$
\boxed{X(U)\cong\int^W\mathcal C(U,W)\times X(W).}
$$
For $a:U'\to U$, 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>.
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