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A useful result due to Linton is that for a cocomplete category $C$ and monad $T$ on $C$, if the category of algebras $C^T$ admits reflexive coequalisers, then it is cocomplete (see here for a sketch ...

Let $\mathcal{V}$ be a monoidal closed (complete, cocomplete, reasonable...) category. Let $\mathsf{T}$ be an enriched monad over $\mathcal{V}$. The forgetful functor $\mathsf{U}: \mathsf{Alg}(\...

In 6.5 of the book by Kelly, Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, No. 10, 2005. the author claims that the $2$-category $\mathsf{Cat}_{\...

Thinking of Cat as a mere 1-category, there is a 1-monad $\Lambda$ for the free cartesian closed category on a category. To every category X it assigns the category $\Lambda(X)$ whose objects are ...