Let $\mathrm{F}: \mathcal{C} \rightleftarrows \mathcal{D} : \mathrm{G} $ be an adjunction with associated monad $\mathrm{T} = \mathrm{G} \mathrm{F} .$

If $\mathcal{D} $ admits coequalizers of $\mathrm{G} $-split pairs, then the comparison functor $ \bar{\mathrm{G}}: \mathcal{D} \to \mathrm{Alg}_{ \mathrm{T}} (\mathcal{C}) $ admits a left adjoint $ \bar{\mathrm{F}}.$

So we obtain an adjunction $ \bar{\mathrm{F}}: \bar{ \mathcal{C} }:= \mathrm{Alg}_{ \mathrm{T}} (\mathcal{C}) \rightleftarrows \mathcal{D} : \bar{\mathrm{G}} $ with associated monad $\bar{\mathrm{T}} = \bar{\mathrm{G}} \bar{\mathrm{F}}.$

If $\mathcal{D} $ admits coequalizers of $\bar{\mathrm{G}} $-split pairs, then the comparison functor $ \mathcal{D} \to \mathrm{Alg}_{ \bar{\mathrm{T}}} (\bar{ \mathcal{C} }) $ admits a left adjoint.

So if we assume that $\mathcal{D} $ admits coequalizers (in fact reflexive coequalizers are enough), we can iterate this process starting with an adjunction $\mathrm{F_1}: \mathcal{C}_1 \rightleftarrows \mathcal{D} : \mathrm{G}_1 $ and get a sequence $ (\mathrm{F_\mathrm{i}}: \mathcal{C}_\mathrm{i}\rightleftarrows \mathcal{D} : \mathrm{G}_\mathrm{i})_{ \mathrm{i} \geq 1 } $ of adjunctions, where $\mathrm{G}_\mathrm{i}: \mathcal{D} \to \mathcal{C}_\mathrm{i}:= \mathrm{Alg}_{ \mathrm{T}_{\mathrm{i}-1} } ( \mathcal{C}_{\mathrm{i}-1} ) $ is the comparison functor for the adjunction $ \mathrm{F_{\mathrm{i}-1}}: \mathcal{C}_{\mathrm{i}-1}\rightleftarrows \mathcal{D} : \mathrm{G}_{\mathrm{i}-1} $ with associated monad $\mathrm{T}_{\mathrm{i}-1} $ if $\mathrm{i} > 1.$

Write $\mathcal{C}_{\infty}$ for the limit of the diagram $ ... \to \mathcal{C}_3 \to \mathcal{C}_2 \to \mathcal{C}_1$ so that we obtain a functor $\mathcal{D} \to \mathcal{C}_{\infty}$.

The functor $\mathcal{C}_{\infty} \to \mathcal{C}_1$ is right adjoint and conservative.

If $\mathcal{D}$ and all the categories $ \mathcal{C}_{\mathrm{i}} $ are presentable, all the right adjoint functors $ \mathrm{G}_{\mathrm{i}} $ preserve small limits and are accessible so that $\mathcal{D} \to \mathcal{C}_{\infty}$ preserves small limits and is accessible and so admits a left adjoint by the adjoint functor theorem.

Does the functor $\mathcal{D} \to \mathcal{C}_{\infty}$ admit a left adjoint without the presentability assumption on $\mathcal{D}?$

Denote $\mathrm{T}_\infty$ the monad associated to the adjunction $\mathcal{C}_{\infty} \rightleftarrows \mathcal{D}.$

It is tempting to believe that the monadic forgetful functor $\mathrm{Alg}_{\mathrm{T_\infty}} ( \mathcal{C}_\infty) \to \mathcal{C}_\infty $ is an equivalence. This is equivalent to the condition that $\mathrm{T_\infty}$ is the identity monad or that the left adjoint of the functor $\mathcal{D} \to \mathcal{C}_\infty $ is fully faithul. Is this true?

What can one say more about $\mathcal{C}_{\infty}$ and the functor $\mathcal{D} \to \mathcal{C}_{\infty}$?