Let $\mathcal{C}$ be any (perhaps small) category. Is there a (co)complete category $\mathcal{C}'$ and an inclusion $\mathcal{C}\hookrightarrow\mathcal{C}'$ which is universal among (co)complete categories containing $\mathcal{C}$? Perhaps something akin to the Yoneda embedding into the category of presheaves over $\mathcal{C}$ is the obvious choice? If not perhaps some other explanation can be given. 



(Co)completions?
The Yoneda $\mathcal{C}\to Psh(\mathcal{C})$ is initial among pairs (D,F) consisting of a cocomplete category $D$ and a functor $F:\mathcal{C}\to D$. That is, it's the initial object of the comma category $\mathcal{C}\downarrow\_{Cat} CoComp$, where $CoComp$ is the category of cocomplete categories with colimit preserving functors between them (technically to form this comma category, we would want to form the comma category of the inclusions of both $\mathcal{C}$ and $\mathcal{CoComp}$ into $Cat$, but, hey, what's an abuse of notation between friends?).


For this reason, we call the category $Psh(\mathcal{C})$ the free cocompletion of $\mathcal{C}$


