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Let $T$ be a monad on an accessible category (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is finitary (i.e. preserves filtered colimits), then the Eilenberg–Moore category of $T$ is also accessible. This is proven (in slightly more generality) in Theorem 2.78 of Adámek–Rosický's Locally presentable and accessible categories, for instance.

We can explicitly describe the subcategory of finitely-presentable objects: they form the full subcategory of the Kleisli category of $T$ spanned by those $A$ that are finitely-presentable in the underlying category. This is proven in the proof of Theorem 6.9 of Bird's Limits in 2-categories of locally-presentable categories (ignoring there the cocompleteness assumption, which is irrelevant here).

Let $T$ be a monad on an accessible category (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is finitary (i.e. preserves filtered colimits), then the Eilenberg–Moore category of $T$ is also accessible. This is proven (in slightly more generality) in Theorem 2.78 of Adámek–Rosický's Locally presentable and accessible categories, for instance.

Let $T$ be a monad on an accessible category (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is finitary (i.e. preserves filtered colimits), then the Eilenberg–Moore category of $T$ is also accessible. This is proven (in slightly more generality) in Theorem 6.9 of Bird's Limits in 2-categories of locally-presentable categories or Theorem 2.78 of Adámek–Rosický's Locally presentable and accessible categories, for instance.

We can explicitly describe the subcategory of finitely-presentable objects: they form the full subcategory of the Kleisli category of $T$ spanned by those $A$ that are finitely-presentable in the underlying category. This is proven in the proof of Theorem 6.9 in Bird's thesis.

Let $T$ be a monad on an accessible category (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is finitary (i.e. preserves filtered colimits), then the Eilenberg–Moore category of $T$ is also accessible. This is proven (in slightly more generality) in Theorem 2.78 of Adámek–Rosický's Locally presentable and accessible categories, for instance.

We can explicitly describe the subcategory of finitely-presentable objects: they form the full subcategory of the Kleisli category of $T$ spanned by those $A$ that are finitely-presentable in the underlying category. This is proven in the proof of Theorem 6.9 of Bird's Limits in 2-categories of locally-presentable categories (ignoring there the cocompleteness assumption, which is irrelevant here).

Let $T$ be a monad on an accessible category (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is finitary (i.e. preserves filtered colimits), then the Eilenberg–Moore category of $T$ is also accessible. This is proven (in slightly more generality) in Theorem 2.78 of Adámek–Rosický's Locally presentable and accessible categories, for instance.

Let $T$ be a monad on an accessible category (i.e. an $\mathbf{Ind}$-category). If the underlying endofunctor of $T$ is finitary (i.e. preserves filtered colimits), then the Eilenberg–Moore category of $T$ is also accessible. This is proven (in slightly more generality) in Theorem 6.9 of Bird's Limits in 2-categories of locally-presentable categories or Theorem 2.78 of Adámek–Rosický's Locally presentable and accessible categories, for instance.

We can explicitly describe the subcategory of finitely-presentable objects: they form the full subcategory of the Kleisli category of $T$ spanned by those $A$ that are finitely-presentable in the underlying category. This is proven in the proof of Theorem 6.9 in Bird's thesis.