If the runtime monoidal category were to be a copy-discard category, there would be a class of morphisms that copy the runtime and use it, but do not return it. After copying the runtime, these would still have a single runtime on the input and a single one on the output. These are read-only morphisms, and they should be modelled apart.
However, premonoidal categories cannot express this property. Read-only morphisms are a blind spot of premonoidal categories.
Tags: premonoidal category.