# Topoi of algebras vs co-algebras, right adjointness compared to left exactness

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.

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?