I'm interested in the fact that every presheaf on a category $C$ is a colimit of representable functors. The [nLab](http://ncatlab.org/nlab/show/co-Yoneda+lemma) claims that this can be stated as:
$$F \simeq \int^{c \in C} Y(c) \otimes F(c)$$
Where $F: C^{op} \to \mathbf{Set}$ and $Y: C \to [C^{op}, \mathbf{Set}]$ is the Yoneda embedding. This can reasonably be called the co-Yoneda Lemma.

The fact that every presheaf can be expressed as a colimit of representables can be proven by shown that:
$$F \simeq \mathrm{colim}\left(\left(\int F\right)^{op} \to C \xrightarrow{Y} [C^{op}, \mathbf{Set}] \right)$$
Here $\int F$ is the [category of elements](http://ncatlab.org/nlab/show/category+of+elements). This is proven for example in Mac Lane, Moerdijk - Sheaves in geometry and logic. My question is:
Is there an easy proof for
$$\int^{c \in C} Y(c) \otimes F(c) \simeq \mathrm{colim}\left(\left(\int F\right)^{op} \to C \xrightarrow{Y} [C^{op}, \mathbf{Set}] \right)$$
The co-Yoneda Lemma and the statement about presheaves seem to be linked by that fact, and the literature suggests (by interchanging the results without mentioning the isomorphism) that this is somehow easy to see, but the only proof I know is by showing the first two identities.