Identity in the Presence of Adjunction

We develop a theory of adjunctions in semigroup categories, that is, 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.