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 $\mathsf{Cat}_{\mathcal{F}}$ of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors is in some sense monoidal closed. Indeed $\mathsf{Cat}_{\mathcal{F}}$ has a very natural notion of internal hom, because $\mathsf{Cat}_{\mathcal{F}}(A,B)$ is $\mathcal{F}$-cocomplete. Kelly's result shows that one can define a tensor $\otimes: \mathsf{Cat}_{\mathcal{F}} \times \mathsf{Cat}_{\mathcal{F}} \to \mathsf{Cat}_{\mathcal{F}}$ having the usual property of a monoidal closed structure (up to replacing isomorphisms with euquivalences of categories.

Q1: Unfortunately, I do not understandwhere $A \otimes B$ is defined, to my understanding he starts mentioning it, but I do not get where the definition is given. Could someone help me to understand it?

After a while, I think I got the definition (even if I do not find it in the book, and I wish someone can tell me what the author is doing there), which is quite involved and comes from a $2$-dimensional adaptation of a classical result in $1$-dimensional category theory. The result is mostly due to *Kock* and *Seal*, but I shall mention the **Chap. 6** of PhD thesis of *Martin Brandenburg*, **Tensor categorical foundations of algebraic geometry**, because he gathered the existing literature in a coherent way.

**Thm.** (Seal, **6.5.1** in Ref.) Let $T$ be a coherent (symmetric) monoidal monad on a (symmetric) monoidal category C. Then $\mathsf{Mod}(T)$ becomes a (symmetric) monoidal category.

Seal does not show that it is monoidal closed, because he does not assume closedness of the base, yet he proves monoidality of the Eilenberg-More category of algebras. Hopefully, if the base is closed, so is the EM-category.

Q2: Unfortunately, to my understanding, the literature contains a plethora of notions ofnicemonads: coherent, commutative, monoidal, strong, Hopf... can someone guide me among these options?Which kind of monads induces a monoidal closed structure on the category of algebras?

Q3: I wanted touseaKock-like resultin the case in which $T$ is the free completion under $\mathcal{F}$-colimits over $\mathsf{Cat}$ in orderto show that $\mathsf{Cat}_{\mathcal{F}} \cong T\text{-}\mathsf{Alg}$ is monoidal closed. Indeed to me, this is what Kelly is secretly doing. Does anyone know areferencethat uses this kind of argument? Indeed here many $2$-dimensional and size-related subtleties should be taken into account.

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