Hi Steve, I study these in quite a bit of detail in my paper Internal 1-topoi in 2-topoi, but I was certainly not the first; for instance, in my Definition 2.14, I cite Street's Cosmoi of internal categories, §9.14, where they are discussed. Sincerely, Joj On Wed, Oct 29, 2025 at 4:32 PM Steven Vickers <s.j.vickers.1@bham.ac.uk<mailto:s.j.vickers.1@bham.ac.uk>> wrote: An nLab page https://ncatlab.org/nlab/show/cartesian+object defines an object of X of a 2-category K to be “cartesian” iff the 1-cell X -> 1 and the diagonal X -> X^2 both have right adjoints. This assumes K has finite 2-products. If, further, K has finite PIE-limits, hence finite powers (cotensors) X^C, then it seems reasonable to define X to be “lex” if every diagonal X -> X^C has a right adjoint. Unless I’ve made a mistake, in the 2-category of categories this characterises the lex categories. Have these lex objects in a 2-category been studied? Steve Vickers. You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files | Leave group | Learn more about Microsoft 365 Groups You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/groups/groupsubscription?source=EscalatedMessage&action=files&smtp=categories%40mq.edu.au&bO=true&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/groups/groupsubscription?source=EscalatedMessage&action=leave&smtp=categories%40mq.edu.au&bO=true&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>