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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!

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The answer is that every enriched presheaf can be expressed as a weighted colimit of representables. The general definition of a weighted colimit goes as follows. Let $\mathcal{A}$ and $\mathcal{C}$ be $\mathcal{V}$-categories, let $F \colon \mathcal{A} \to \mathcal{C}$ be a $\mathcal{V}$-functor (the "diagram"), and let $W \colon \mathcal{A}^\mathrm{op} \to \mathcal{V}$ be a $\mathcal{V}$-functor (the "weight"). A colimit of $F$ weighted by $W$ is an object $W \ast F$ of $\mathcal{C}$ together with a $\mathcal{V}$-natural isomorphism $$\mathcal{C}(W\ast F,C) \cong [\mathcal{A}^\mathrm{op},\mathcal{V}]\left(W,\mathcal{C}(F-,C)\right),$$ where $[\mathcal{A}^\mathrm{op},\mathcal{V}]$ denotes the $\mathcal{V}$-category of $\mathcal{V}$-enriched presheaves on $\mathcal{A}$.

To turn to your question, let $\mathcal{A}$ be a small $\mathcal{V}$-category and let $W \colon \mathcal{A}^\mathrm{op} \to \mathcal{V}$ be a $\mathcal{V}$-functor (a "$\mathcal{V}$-enriched presheaf"). It follows from the (strong) enriched Yoneda lemma that $W$ is the colimit of the Yoneda functor $Y \colon \mathcal{A} \to [\mathcal{A}^\mathrm{op},\mathcal{V}]$ weighted by $W$; i.e. $W \cong W\ast Y$.

When $\mathcal{V}=Set$, there is a general formula for expressing a weighted colimit as an ordinary "conical" colimit over the category of elements of the weight, which in the above case reproduces the formula you give in your question.

For a reference, see around display (3.17) of Kelly's Basic concepts of enriched category theory (that's page 41 of the TAC reprint) or Theorem 6.6.18 of volume 2 of Borceux's Handbook of categorical algebra. You may also wish to read Chapter 7 of Riehl's Categorical Homotopy Theory for a treatment of weighted colimits.

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