  We develop a theory of adjunctions in semigroup categories, i.e. monoidal
categories without a unit object. We show that a rigid semigroup category is
promonoidal, and thus one can naturally adjoin a unit object to it. This
extends the previous results of Houston in the symmetric case, and addresses a
question of his. It also extends the results in the non-symmetric case with
additional finiteness assumptions, obtained by Benson-Etingof-Ostrik,
Coulembier and Ko-Mazorchuk-Zhang. We give an interpretation of these results
using comonad cohomology, and, in the absence of finiteness conditions, using
enriched traces of monoidal categories. As an application of our results, we
give a characterization of finite tensor categories in terms of the finitary
2-representation theory of Mazorchuk-Miemietz.
