It is well known that, for a small category $\mathbf A$, the category $\widehat{\mathbf A} = [\mathbf A^\circ, \mathbf{Set}]$ of presheaves on $\mathbf A$ together with the Yoneda embedding $\mathbf A \to \widehat{\mathbf A}$ exhibits $\widehat{\mathbf A}$ as the cocompletion of $\mathbf A$ under small colimits.

Where was this first observed? If an observation first appears independently of a proof, I would be interested in knowing where a proof first appears too.

Bunge's 1966 thesis *Categories of Set-Valued Functors* seems a likely candidate, as it is concerned with properties of presheaf categories. However, I have been unable to obtain a copy of the thesis, so I do not know whether it appears here, and it is not mentioned in Bunge's summary of thesis, *Regular categories*.

isthe Kan extension along the Yoneda embedding). Maybe it is even in the last sections of Kan'sAdjoint functorsfrom 1958 (where he defines limits, adjoints and, well, Kan extensions - it all started there...), which I skimmed through, but I doubt it. $\endgroup$