The fact that a category is a presheaves category if and only if it is cocomplete, regular and atomic is due to Bunge but I cannot find the proof anywhere, does anybody know where it appears?
Side question: is there a characterization of cocomplete atomic cartesian closed categories?
This is proven in the enriched context in Theorem 4.16 of Bunge's Relative Functor Categories and Categories of Algebras (note that she proves the dual result), and specialised to $\mathcal V = \mathbf{Set}$ in Corollary 4.19 ibid. Note, however, that her characterisation also requires well-poweredness and co-well-poweredness.