Are monoidal semi-categories well-defined? Categories without identities are called Semicategories  or Semigroupoids (https://ncatlab.org/nlab/show/semicategory).
Is there a definition for a monoidal category without identities (e.g. a monoidal semicategory)? Or do the axioms for a monoidal category break when we remove identities? If they do break, are there special cases where the monoidal structure would still be well-defined (e.g. cartesian monoidal)?
 A: If you don't have identity morphisms, then this won't work. Firstly, because you don't have identities, you cannot state the pentagonal diagram or any of the triangular diagrams which come with the definition of a monoidal category. This is because they involve identities. Now it's not a big deal to lose the triangular diagrams--if you lost them you'll get semi-monoidal categories which I said in my previous answer is in general coherent--but losing the pentagonal diagram is absolutely the worst thing imaginable in a monoidal category. It means you can't even coherently discuss the tensored product of 4 elements, much less high tensor products, so you no longer have a monoidal category.
The philosophy behind a monoidal category is that given any tensor product, there needs to be a canonical isomorphism between any two ways to paranthesize that tensor product (i.e. there needs to be coherence). And in a monoidal category, most of those canonical isomophisms involve identities (boxed in with associators and unitors), and there's just no way around that. So identity morphisms are extremely necessary in the definition of a monoidal category, and in fact are what allow you (along with associators and unitors) to discuss canonical isomorphisms between any two ways of paranthesizing a tensor product. If you lose identities, you cannot achieve the philosophy of a monoidal category even in some reduced sense.
