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Fix a regular cardinal $\kappa$ and let $\mathcal{C}$ be a $\kappa$-presentable $\infty$-category (comments about the 1-categorical case are welcome as well!).

I'm looking for a reference for the following statement (which in particular I hope is true).

There exists a ("canonical") adjunction of $\infty$-categories

$$ Alg_{(-)}(\mathcal{C}): Monad_{\kappa}(\mathcal{C}) \rightleftharpoons Pr^{L,\kappa}_{\mathcal{C} /}:T_{(-)} $$

Where on the left we have $\kappa$-accessible monads on $\mathcal{C}$ and on the right $\kappa$-presentable categories equipped with a left adjoint (whose right adjoint is $\kappa$-accessible of course) from $\mathcal{C}$.

The right adjoint of the adjunction should send a left adjoint functor to it's corresponding monad on $\mathcal{C}$ and the left adjoint of the adjunction should send a monad to it's category of algebras equipped with the free algebra functor from $\mathcal{C}$.

Similarly (in fact more pressingly) I would like to know if there's also a dual statement for an adjunction of the form:

$$S_{(-)} : Pr^{L, \kappa}_{/ \mathcal{C}} \rightleftharpoons CoMonad_{\kappa}(\mathcal{C}) :CoAlg_{(-)}(\mathcal{C})$$

Aside: The reason I phrased the second adjunction in a more suspicious tone is I do not even know a reference for the statement that the category of coalgebras over a $\kappa$-accesible comonad on a $\kappa$-presentable category is $\kappa$-presentable (I am not even sure whether or not this should be true).

EDIT: I just realized that the statement for coalgebras with $\kappa = \omega$ can't be true. Indeed there are many counterexamples in categories of comodules. In light of that I could change the question to be about $\kappa > \omega$ or just remove the cardinal restriction all together. Since no answers were given yet I think it's better to just leave the question as is.

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    $\begingroup$ I recently worked on the exact same question and I couldn't find a reference. For the first question I needed it in a paper, and it does not seem to be treated anywhere, so I'm currently writting a proof. The second question I recently asked a reference request for the 1-categorical case of your aside on MO as mathoverflow.net/questions/350351/… and it didn't get an answer. I have written a proof myself that the category of co-algebra is $\kappa$-presentable when $\kappa$ is uncountable (I don't think it is true for $\kappa=\omega$), but... $\endgroup$
    – Simon Henry
    Commented Jan 23, 2020 at 14:21
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    $\begingroup$ It will be difficult to upgrade this proof to the $\infty$-categorical case (it is fairly technical...) $\endgroup$
    – Simon Henry
    Commented Jan 23, 2020 at 14:22
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    $\begingroup$ In the 1-categorical case, a result close to what you want is Theorem 6.5 of tac.mta.ca/tac/volumes/28/13/28-13abs.html, which is due to Dubuc (see Proposition 6.2 there). $\endgroup$
    – Tom Leinster
    Commented Jan 23, 2020 at 15:10
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    $\begingroup$ Oh I see. In that case, maybe my reference to Dubuc's theorem is less relevant. Not sure. What confused me was the part of your question where you wrote "categories with left adjoints". Since a category can't have a left adjoint (!), I assumed that was an error, and that you'd meant to say that the functor from your category to C had a left adjoint. But now I guess what you meant is that the objects of your right-hand category are categories, and the maps in it are the functors that are left adjoints (i.e. have right adjoints). $\endgroup$
    – Tom Leinster
    Commented Jan 23, 2020 at 17:09
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    $\begingroup$ @MikeShulman "... frequently turns out not to be the case in locally-presentable-category theory." What are some good examples demonstrating this? $\endgroup$
    – Saal Hardali
    Commented Jan 25, 2020 at 16:25

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