nLab
free completion

For a small category $S$, just as the presheaf category ${\mathrm{Set}}^{S^{\mathrm{op}}}$ is the free cocompletion of $S$, by duality we can conclude that $({\mathrm{Set}}^S{)}^{\mathrm{op}}$ is the free completion of $S$. This means that any functor $S\to C$ where $C$ is complete factors uniquely (up to isomorphism) through the “dual Yoneda embedding” $S\to ({\mathrm{Set}}^S{)}^{\mathrm{op}}$ via a continuous functor $({\mathrm{Set}}^S{)}^{\mathrm{op}}\to C$.

This operation is a 2-monad which is lax idempotent but not (even weakly) idempotent. In analogy with the general concept of completion, we might call the operation of any lax idempotent monad on a $2$-category a ‘free completion’. See discussion at completion.