The paper in question never appeared; see list of category theory papers that never appeared.

This is a different paper: both "Categories with structure – biadjoints for algebraic functors" and "Doctrinal adjunction" are referenced separately in Basic Concepts of Enriched Category Theory.

Perhaps this should be a community wiki answer instead?

Could you give a more precise reference (e.g. theorem or page number) to your book?

Wonderful, thank you so much!

Do you have a copy of "Bialgebras in locally presentable categories" you might be able to share? I have previously looked for the preprint without success.

It would be nice to give some references for these facts.

@JonasFrey: okay, great. It's good to make the distinction, so thanks for pointing it out!

However, I think the term "enrichment" suggests the first perspective, so I think it would be fair to say there is no concept of "enrichment of a semicategory over a semimonoidal category". There is just the concept of "enriched semicategory". Most likely you are pointing out the existing wording in my answer is misleading in this respect? I will tweak the wording accordingly, thank you. (3/3)

Enrichment in semigroupal/semimonoidal categories is motivated by the second perspective and, here, it turns out that there is no notion of underlying semicategory. So the first perspective on enrichment does not generalise to enrichment in semimonoidal categories. However, one can find examples of such enriched semicategories (as in the linked paper), which suggests the definition is reasonable. (2/3)

@JonasFrey: there are multiple ways to view the concept of enriched category. One is as structure on a category, in which case one expects to be able to recover the underlying category from the enriched category. However, another perspective is simply a category-like structure whose homs form not a set, but the object of some other category. Here, a priori, there is no reason to expect this notion of enriched category to have an underlying category, though it turns out to be true by homming out of the unit. (1/3)

Does this question answer your question?

You don't need to present the monoidal structure via the indexing: I just mentioned that as another way to see the structure. The easiest way to present the monoidal structure is abstractly, as I did in the first paragraph.

This is definitely a well known construction (e.g. see this nLab entry), but I'm afraid I don't know a reference in the literature for it...