Every enriched functor into the base is a weighted colimit of representables when the displayed weighted colimit exists

Evaluate the displayed coend at an object $K$ of $\mathcal{A}$. By [L2] and [L3], its value is $C_K:={\int }^{A}F(A)\otimes \mathcal{A}(A,K)\mathrm{.}$

Fix $X\in \mathcal{V}$ and define a $\mathcal{V}$-functor $G_X:{\mathcal{A}}^{\mathrm{o}\mathrm{p}}\to \mathcal{V}$ by $G_X(A)=[F(A),X]$. The coend universal property and the closed structure give natural isomorphisms $[C_K,X]\cong {\int }_A[F(A)\otimes \mathcal{A}(A,K),X]\cong {\int }_A[\mathcal{A}(A,K),[F(A),X]]\mathrm{.}$ In ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}$ one has ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}(K,A)=\mathcal{A}(A,K)$, so applying [L1] to $G_X$ identifies the last end with $G_X(K)=[F(K),X]\mathrm{.}$

The isomorphism $[C_K,X]\cong [F(K),X]$ from step 2.1 is natural in $X$. The enriched Yoneda principle therefore gives $C_K\cong F(K)$. These isomorphisms are natural in $K$, so they assemble into an isomorphism of $\mathcal{V}$-functors between the displayed weighted colimit and $F$.