Suppose that $\mathcal C$ is an ordinary category and $F:\mathcal C^{op}\longrightarrow Set$ a functor. Then, we can form the category $\mathcal C/F$ as follows : each object is a morphism of functors from $h_X\longrightarrow F$, where $X\in \mathcal C$ and $h_X=Hom(\_\_,X):\mathcal C^{op}\longrightarrow Set$ is the representable functor.

In other words, we are looking at the category of pairs $(X,x)$, where $X\in \mathcal C$ and $x\in F(X)$. This is clear from Yoneda lemma.

Then, we can write down the functor $F$ as a colimit :

$$F=\underset{(X,x)\in \mathcal C/F}{colim}\textrm{ }h_X $$

My question is whether we have a version of this for enriched categories. In other words, suppose that $\mathfrak D$ is a category enriched over a closed symmetric monoidal category $(\mathcal V,\otimes,1)$.

Let $\mathcal F:\mathfrak D^{op}\longrightarrow \mathcal V$ be an enriched functor. For each $X\in \mathfrak D$, we already have a representable functor:

$$H_X: \mathfrak D^{op}\longrightarrow \mathcal V\qquad H_X(Y):=\mathfrak D(Y,X)$$ for any $Y\in \mathfrak D$.

Can we construct a category $\mathfrak D/\mathcal F$ in this case and express $\mathcal F$ itself as a colimit of the form:

$$\mathcal F = \underset{??}{colim}\textrm{ }H_X$$

Of course the key must be an enriched form of Yoneda lemma. In the case of enriched categories, there are 2 forms of Yoneda lemma, the weak form and the strong form. I would prefer if the answer can be given with the help of the weak form.

Of course it would be great if there is a reference where this formula is clearly explained.

Thanks!