Given a topos $E$, the subobject classifier $Ω_E$ is known to be the initial internal frame. On the other hand, the unique frame morphism $i:{Ω_E}\rightarrow{H}$ is known to have a right adjoint $τ$ as the classifying map of the top over $H$ (see Topos Semantics for Higher-Order Modal Logic. S Awodey, K Kishida, HC Kotzsch. arXiv:1403.0020). The resulting comonad $i\circτ$ models the S4 system of Modal Logic.

Is this comonad idempotent as in previous works (A topos-theoretic approach to reference and modality. Reyes, Gonzalo E. Notre Dame J. Formal Logic 32 (1991), no. 3, 359--391)?

In general: which are the conditions for a comonad arising form an adjunction be idempotent (I am looking for conditions beyond the case of the base category being a poset)?


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