 Free cocompletion

For a small category $\mathcal{C}$, its presheaf category is the free enlargement under small colimits. Each categorical presheaf $P$ is the colimit of the diagram sending $(C,x)\in \int P$ to the representable presheaf $yC$. The map at $(C,x)$ sends $f:A\to C$ to $P(f)x$. It is surjective at each $A$ because $z\in P(A)$ is the image of $1_A$ at $(A,z)$. If two representatives have the same image $z$, they are both identified with the representative $1_A$ at $(A,z)$ by arrows in the category of elements, proving injectivity. Thus $P\cong {\mathrm{colim}}_{(C,x)\in \int P}yC$. For a cocomplete target, a functor on $\mathcal{C}$ extends by these colimits, uniquely up to natural isomorphism among colimit-preserving extensions.