“Elgot categories” is a term charged with multiple incompatible meanings.

I have employed “imperative categories” for Freyd categories with strong Elgot structure---in the sense of Ghalayini and Krishnaswami---“imperative monoidal categories” for Elgot categories in the sense of Nester (and, while not necessarily asking for a natural numbers object, certainly expecting it in most examples), and “locally-posetal imperative categories” for those we employ for program logics reasoning.