Semi-adjunction are a notion of adjunctions between semicategories, though it is often interesting to consider semi-adjunctions between ordinary categories.
Susumu Hayashi, “Adjunction of semifunctors: categorical structures in nonextensional lambda calculus.” Theoretical computer science 41 (1985): 95-104.
Raymond Hoofman, “A note on semi-adjunctions.”, 1990, (pdf)
Raymond Hoofman, “The theory of semi-functors.” Mathematical Structures in Computer Science 3.1 (1993): 93-128.
Raymond Hoofman and Ieke Moerdijk, “A remark on the theory of semi-functors.” Mathematical Structures in Computer Science 5.1 (1995): 1-8.
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