The Yoneda $\mathcal{C}\to Psh(\mathcal{C})$ is initial among pairs of functors $\mathcal{C}\to D$ with $D$ cocomplete. 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 $\mathcal{C}$

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 $\mathcal{C}$