The 2-category AdjAdj is the free adjunction (walking adjunction).

A 2-functor AdjKAdj \to K is an adjunction in the 2-category KK. These 2-functors form one version of the 2-category of adjunctions of KK.


AdjAdj is the 2-category freely generated by

  • two objects: aa and bb,

  • two morphisms: L:abL: a \to b and R:baR: b \to a,

  • and two 2-morphisms, called the “unit” and “counit”: i:1 aLRi: 1_a \to L R and e:RL1 be: R L \to 1_b, satisfying two relations, called the “triangle equations”.

The restrictions of the free adjunction, AdjAdj, to the sub-2-categories spanned by one endpoint, aa, or the other, bb, define the free monad and the free comonad.


  • C. Auderset, Adjonction et monade au niveau des 2-categories, Cahiers de Top. et Géom. Diff. XV-1 (1974), 3-20.

  • John Baez, This Week’s Finds in Mathematical Physics (Week 174), (TWF174)

  • Kevin Coulembier, Ross Street, Michel van den Bergh, Freely adjoining monoidal duals, arXiv:2004.09697 (2020). (abstract)

  • Dieter Pumplün, Eine Bemerkung über Monaden und adjungierte Funktoren, Math. Annalen 185 (1970), 329-377.

  • Stephen Schanuel and Ross Street, The free adjunction, Cah. Top. Géom. Diff. 27 (1986), 81-83. (numdam)

Last revised on June 16, 2020 at 06:52:31. See the history of this page for a list of all contributions to it.