Total Category

A category $\mathcal{C}$ is total if the Yoneda Embedding $y:\mathcal{C}\to\hat{\mathcal{C}}$ has a Left Adjoint.

This imposes some rather strong Cocompleteness conditions on $\mathcal{C}$, as Left Adjoints Preserve Colimits. This not only means that $\mathcal{C}$ is cocomplete, but also that we know how to compute said Colimits.