Is there anything known about what happens when you create the free almost-SMCC from a set of objects (following perhaps the construction of Troelstra 1992:81-86, but leave out the unit axiom arrows ($\lambda_A:1*A\rightarrow A$)? I can't seem to find anything out about such things in the literature.
The motivation is that there is an activity in linguistics called `glue semantics' in which you use 'rudimentary' linear logic a.k.a. free SMCCs to model assembly of word meanings, but the unit and unitors aren't used in it at all, and there is a bit of an issue in handling the meanings of the actual words, which such a category might help with.
Further Remarks based on the useful comments below: for the linguistic application, it might even be good to have arrows such as $1\rightarrow[A,A]$ and the one discussed in the Houston et al. paper below be missing, because in meaning assembly we want to go from postulates representing word meanings to an assembled meaning for the whole. E.g. Walk$\otimes$John$\rightarrow$Walk(John) where Walk is of type $[E,P]$ and John is of type $E$, and we don't want the conclusion to be filled up with contentless extra junk such as $[A,A]$ for arbitrary $A$.
This problem is currently avoided by stating the type that you want the assembled meaning to have, typically `proposition', but this is oversimplistic, and it would be nice to have an assembly scheme that would list the finite number of sensible assemblies with no unwanted junk. Basically, a logic where the only thing that follows from nothing is nothing. So the 'naive solution' is perhaps not the wrong one here.