A particular case of this is in SGA4, IV 9.5.4, where it is shown that the category obtained by (Artin) glueing along a finite-limit- preserving functor between Grothendieck toposes is a Grothendieck topos iff the functor is accessible. It was Gavin Wraith, in JPAA 4 (1974), who first observed that Artin glueing is a particular case of forming a category of coalgebras (and therefore works for elementary toposes without the accessibility condition). Peter Johnstone On Thu, 9 May 2013, David Roberts wrote:

Hi all,

I am just wondering where it was first stated (for both directions) that the category of coalgebras for a comonad on a Grothendieck topos E is again Grothendieck if and only if the underlying endofunctor of E is accessible.

A modern argument might go as: the topos of coalgebras is Grothendieck if and only if it is locally presentable if and only if the endofunctor is accessible, the original probably just mentioned preservation of filtered colimits.

Many thanks,

David Roberts

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