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May 19, 2018 at 18:06 comment added Tim Campion If the exactness conditions are obvious, then this really reduces to the fact (due to Bird) that locally presentable categories (and left adjoint functors) are closed under pseudo-limits in $Cat$. And indeed, in that case also most of the work goes into showing that one has a generating set.
Mar 28, 2011 at 18:25 vote accept Todd Trimble
Mar 28, 2011 at 18:16 history answered Mike Shulman CC BY-SA 2.5