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It is easy to see that fixing the cartesian arrow also fixes the vertical arrow. If everybody is happy with this solution I am fine. It is in the same spiri= t as chosen pullbacks or chosen finite limits. The reason for my reluctance is that nobody would consider as a natural notion a vector space with a chosen basis. The virtue of strong choice is that then such additional structure can be pulled out of the hat at demand. 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