# nLab free completion

For a small category $S$, just as the presheaf category $Set^{S^{op}}$ is the free cocompletion of $S$, by duality we can conclude that $(Set^S)^{op}$ 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” $S\to (Set^S)^{op}$ via a continuous functor $(Set^S)^{op}\to C$.

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

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