There are at least two standard ways of unitizing a (small) semigroup $\mathbb A$:
(i) We add an identity regardless that $\mathbb A$ is already unital.
(ii) We add an identity only if none is already available.
In the former case, the unitization process is functorial, as it amounts to the existence of a left adjoint to the canonical forgetful functor from the category of small categories to the category of small semicategories (in the sense of B. Mitchell).
Question. Is there any standard terminology to differentiate (i) from (ii)? I would be content with something like "(i) is occasionally called the unitization à la X" or "(ii) is referred to by some authors as Y's unitization".
Thanks in advance.