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I added a remark after “Rafael’s theorem”. By the way, the equations obeyed by ν and ζ 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⊣G be a pair of adjoint functors. Then F is separable iff the unit η:1→GF has a section (= a natural transformation ν which is its right inverse, ν∘η=1). G is separable iff the counit ε:FG→1 has a retraction (i.e., a natural transformation ζ that is its left inverse, η∘ζ=1).
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