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At comma object, Eduardo Pareja-Tobes put a query box which does not seem to have been ‘answered’. This was some time ago it seems.
Thanks! I’ve replied and suggested to continue here.
This same person also wrote for us factorization category and twisted arrow category.
Six years later, I’m copying the conversation here and deleting the query box:
Eduardo Pareja-Tobes: Not sure about this but, with the strict definition I think you end up having specified isos all around at the level of morphisms; comma categories as normally defined are comma objects in Cat, but not strict ones (of course they’re equivalent to the strict ones). I remember reading something like this in Makkai-Paré Accessible categories book
Mike Shulman: As far as I can tell, they are strict. Given $D$, functors $p':D\to A$, $q':D\to B$ and a natural transformation $\sigma:f p'\Rightarrow g q'$, these data specify exactly for every $d\in D$, a triple $(p'(d), q'(d), \sigma_d)$ which is an object of the comma category. Perhaps you are remembering a related remark about pseudo-pullbacks versus iso-comma objects? (If you post your comments at the nForum, for instance on this discussion, other people will be more likely to see it.)
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