# Reference for the alternative construction of the free cocompletion

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?

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.