Let $\mathcal{V}$ be a semi-monoidal category, meaning it satisfies the axioms of a monoidal category except missing a unit and the unit axiom. One could then still go about defining a $\mathcal{V}$-category by dropping the requirement of having unit morphisms.


One concern is that without unit morphisms there is no way to define an underlying category $\mathcal{C}\_0$ associated to a $\mathcal{V}$-category $\mathcal{C}$. However other than that it seems to me that other parts of the theory makes sense.


Has such categories been considered previously anywhere in literature? I can find several mentions of semi-monoidal category e.g. <https://arxiv.org/abs/math/0507349>
but no mentions of semi-enriched categories. Of particular interest to me would be if there is a version of the enriched Yoneda lemma for such categories.



Enriched categories over a semi-monoidal category
Expanding upon my comment: categories without units are called [semicategories](https://ncatlab.org/nlab/show/semicategory). You can enrich a semicategory in a semigroupal category, which is what you describe. The Yoneda lemma is subtle with semicategories, but see [On regular presheaves and regular semi-categories](http://www.numdam.org/item/CTGDC_2002__43_3_163_0/). However, note that the authors work with semicategories enriched in a monoidal category: this is because, despite the definition of enriched semicategory and semifunctor not needing a unit in $\mathcal V$, a unit is necessary to define an enriched notion of natural transformation between semifunctors. I am not aware of a reference that explicitly develops the theory of semicategories enriched in semigroupal categories.


