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Given any small category $\mathcal{C}$, the Yoneda embedding $y:\mathcal{C} \to \mathbf{Set}^{\mathcal{C}^{\operatorname{op}}}$ is well-known to represent the free cocompletion of $\mathcal{C}$. That is, the functor $F \mapsto F \circ y$ gives an equivalence of categories between cocontinuous functors from $\mathbf{Set}^{\mathcal{C}^{\operatorname{op}}}$ to $\mathcal{D}$ and functors from $\mathcal{C}$ to $\mathcal{D}$ for any cocomplete category $\mathcal{D}$.

But there is also an alternative construction which seems more intuitive and obvious.

Namely, one could define the free cocompletion $\hat{\mathcal{C}}$ as follows:

  • The objects of $\hat{\mathcal{C}}$ are the small diagrams in $\mathcal{C}$ (i.e., small categories $I$ with a functor $D:I \to \mathcal{C}$).
  • The set of morphisms from $(I, D)$ to $(J, E)$ in $\hat{\mathcal{C}}$ is given by a "limit of colimits" in the category of sets. To be precise, it is $\operatorname{lim}_{i \in I} \operatorname{colim}_{j \in J} \operatorname{Hom}_{\mathcal{C}}(D(i), E(j))$.
  • Composition and identities in $\hat{\mathcal{C}}$ are defined using the ones in $\mathcal{C}$.

The functor $\mathcal{C} \to \hat{\mathcal{C}}$ would then map each object $X$ to the corresponding "singleton" diagram $(1, X)$ (where $1$ is the category with only a single object and its identity morphism and $X$ is treated as a functor $1 \to \mathcal{C}$ in the obvious way, also denoted $X$ by abuse of notation). Also, each object $(I, D)$ of $\hat{\mathcal{C}}$ should formally represent the colimit of $D$.

The cocontinuous extension of any functor $F:\mathcal{C} \to \mathcal{D}$ would then map $(I, D)$ to $\operatorname{colim}_{i \in I} F(D(i))$.

So, is there any source on category theory that mentions the above alternative construction of the free cocompletion?

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This is essentially the topic of Beurier–Pastor–Guitart's Presentations of clusters and strict free-cocompletions. They define a category $\mathrm{Clu}(\mathscr C)$ for any category ($\mathscr C$ is not necessarily small), and prove (Theorem 4.4) that it exhibits the free strict cocompletion of $\mathscr C$. The morphisms from a diagram $P$ to a diagram $Q$ in $\mathscr C$ is defined to be the set of clusters from $P$ to $Q$ (Definition 3.1): they prove this is equivalent to the limit–colimit formula you give in Proposition 3.11.

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