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I also said that for most weak equivalences you meet the pseudoinverse can be constructed. But for establishing this in general you need global choice. Just look at the proof in MacLane or similar textbooks. Or where do they prove that the assumption of a pseudo inverse is a conservative extension without having a pseudo inverse at hand? I do not believe that big choice is needed in down to earth mathematics. I agree when applying general results to particular situations these general principles can be mostly (always? but why?) eliminated. But I do not know of any argument that for all functors between Grothendieck toposes preserving finite limits and small colimits entails the existence of a right adjoint. These (unintentionally) strong principle have been introduced just for convenience and later it has been found out that they have suprising consequences. Thomas ---------- You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. Leave group: https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.as= hx?source=3DEscalatedMessage&action=3Dleave&GuestId=3D4eb9b40c-9b3a-48a5-97= 81-836e5a171e8b