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The recent series of papers by Lack and Tendas are good references for enriched accessible categories: Flat vs. filtered colimits in the enriched context (Lack–Tendas) On continuity of accessible fun …

In Theorem 5.19 of Kelly's Basic Concepts of Enriched Category Theory, it is proven that a fully faithful functor $K \colon \mathcal A \to \mathcal C$ is dense if and only if $\mathcal C$ is the closu …

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 …

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 …

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 $\mathcal C$, …

Most Kan extensions arising in nature are pointwise, and this observation prompts Kelly to write [1]: Our present choice of nomenclature is based on our failure to find a single instance where a [non …

Expanding upon my comment: categories without units are called semicategories. You define a notion of semicategory enriched in a semigroupal category, which is what you describe. The Yoneda lemma is s …

This is exactly the subject of the paper Coends of higher arity by Loregian and de Oliveira Santos.

In an email to the categories mailing list dated 21 August 2003, Street writes: Max reminded me of his old result (not in the LaJolla Proceedings, but known soon after) that a monoidal V-category is …

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 …

By a result of Foltz, and Freyd and Street, a category $C$ is essentially small (i.e. equivalent to a small category) if and only if both $C$ and $[C^{\text{op}}, \mathrm{Set}]$ are locally small. I a …

Internal categories and enriched categories can both be realised as monads in certain bicategories. If $\mathcal E$ is a category with pullbacks, then a monad in $\mathbf{Span}(\mathcal E)$ is a categ …

Let $\mathscr K$ be a small 2-category. It follows from $\mathrm{Cat}$-enriched category theory that the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors is given by the 2-categ …

I think it worth mentioning that precisely the notion described in the question is given in Cockett–Koslowski–Seely's Morphisms and modules for poly-bicategories, where it is called a multi-bicategory …