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I’ve disambiguated links to cartesian category. I suggest that we avoid this term.
Thanks. Okay, so I am in 2:1 minority about this.
The problem with “cartesian” for “finitely complete” is that it is also used to refer only to finite products, as for instance in cartesian bicategory and cartesian object. I find it ironic that elsewhere in the Elephant, Johnstone declines to use the absolutely standard word “cartesian” for cartesian morphisms on the grounds that the word “cartesian” has been “rather overworked by category-theorists” – but he does choose to use “cartesian” for “finitely complete”, where it is not standard at all. I think “finitely complete” has the advantage of being completely unambiguous. If it seems too long, then one can also just say “lex”, which is also completely unambiguous, but may require explanation to people who haven’t heard it before.
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