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I added a remark after “Rafael’s theorem”. By the way, the equations obeyed by $\nu$ and $\zeta$ don’t seem to match the verbal descriptions of them as right and left inverses in the statement of Rafael’s theorem, so I hope someone can sort this out and fix this:
Rafael’s theorem. Let $F\dashv G$ be a pair of adjoint functors. Then $F$ is separable iff the unit $\eta:1\to G F$ has a section (= a natural transformation $\nu$ which is its right inverse, $\nu\circ\eta = 1$). $G$ is separable iff the counit $\epsilon:F G \to 1$ has a retraction (i.e., a natural transformation $\zeta$ that is its left inverse, $\eta\circ\zeta =1$).
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