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I added a remark after “Rafael’s theorem”. By the way, the equations obeyed by and 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 be a pair of adjoint functors. Then is separable iff the unit has a section (= a natural transformation which is its right inverse, ). is separable iff the counit has a retraction (i.e., a natural transformation that is its left inverse, ).
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