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.

the canonical unitization. – Aaron Meyerowitz Jan 16 '13 at 16:11forced unitizationand (ii) has been called theconditonal unitization-- I think I have seen this terminology in the pink book of Helemskii, for instance, but my memory is rusty. – Yemon Choi Jan 16 '13 at 19:49