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I am asking about topoi of coalgebras over a comonad and algebras over a monad. The comonad statement I am aware of is as follows:

Let $\mathcal{E}$ be a topos. Then if a comonad $T \colon \mathcal{E} \rightarrow \mathcal{E}$ is left exact, then the category of coalgebras $\mathrm{TCoAlg}(\mathcal{E})$ is itself a topos.

While the monad version is:

Let $\mathcal{E}$ be a topos. Then if a monad $T \colon \mathcal{E} \rightarrow \mathcal{E}$ has a right adjoint, then the category of algebras $\mathrm{TAlg}(\mathcal{E})$ is itself a topos.

Now it seems to me, that the proper dual should be talking about right exactness only for monads instead of having a right adjoint. So in comonad version, left exact here means $T$ preserves finite limits. In monad version, by having a right adjoint (being a left adjoint) $T$ preserves all colimits.

Can $T$ be weakened to right exact one in the statement for algebras or is there some subtle dualization reason why we suddenly talk about preserving all colimits, not just the finite ones?

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    $\begingroup$ If you dualised everything, you would be asking instead about monads on cotoposes, which is why the statement for monads is different to that for comonads. $\endgroup$
    – varkor
    Jan 23 at 20:04
  • $\begingroup$ @varkor that indeed seems to be my mistake here $\endgroup$
    – Nift
    Jan 23 at 20:22

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