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But philosophical issues can be relevant and in this case definitely are. I think the following analogy might be helpful. Using strong choice principles every vector space can be endowed with a basis. But nobody thinks that they should form part of the definition or even be preserved by morphisms between them. And so it is with Grothendieck fibrations. They are the right notion but one may choose cleavages. However, they are not part of the structure and certainly should not be preserved by morphism, i.e. cartesian functors. 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