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There are a few things going on here. In short, what you are describing is a pseudodistributive law between a lax-idempotent relative pseudomonad and a pseudofunctor. The presheaf construction, as you …

A PDF of Freyd's 1960 thesis Functor Theory is available in the Category Theory Archive.

In Comprehensive factorisation systems, Berger and Kaufmann establish a correspondence between certain orthogonal factorisation systems (the motivating example being the discrete fibration/final funct …

Hyland's Elements of a theory of algebraic theories describes a precise connection between multicategories and $\mathbf{Prof}$ in Section 4.3. I shall briefly describe the intuition; a complete pictur …

Assuming $\mathcal C$ and $\mathcal D$ are locally small and $\mathcal C$ is small-cocomplete, a functor $F \colon \mathcal C \to \mathcal D$ is a left adjoint if and only if it is small-cocontinuous …

It is certainly the case that the duals of internal homs have appeared significantly less in the categorical literature. I've included a few more references below, but I am not sure this is a satisfyi …

In "Partial Horn logic and cartesian categories", E. Palmgren and S. J. Vickers state without proof that "The theory of categories is not algebraic." Is there a reference, or an elementary argument, f …

Let $C$ be a category. There is a full subcategory $\text{IdemMnd}(C) \hookrightarrow \text{Mnd}(C)$ of the category of monads on $C$ spanned by the idempotent monads. Given a monad $T$ on $C$, suppos …

The creation of weighted limits by the forgetful functor from the $\mathscr V$-category of algebras for an enriched (relative) monad is proven in Proposition 2.5 of Arkor–McDermott's Relative monadic …

The general problem of giving a categorical construction of the free category with finite coproducts and products (or "free sum–product category") seems to still be open, though there are several work …

There is a standard notion of a bicategory enriched in a monoidal bicategory, which is a categorification of the notion of category enriched in a monoidal category. The standard reference would be Gar …

This follows from Lemma 2.3 and Proposition 3.2 of Creurer–Marmolejo–Vitale's Beck's theorem for pseudo-monads, together with fact that the bicategorical Yoneda embedding preserves bilimits. Presumab …

This is a special case of the general construction of cocompletions that preserve existing colimits. The general statement can be found as Theorem 6.23 of Kelly's Basic Concepts of Enriched Category T …

An answer has already been accepted, but I believe it does not really answer the question as given, so here is another. You can define $\mathcal V$-enrichment as structure on a category $\mathcal C$, …

A relevant reference is §3 of Kelly's Doctrinal adjunction: in particular, see Theorem 3.1, which states that, if the adjunction is reflective, then the subcategory inherits monoidal structure if and …