Dual pair in a monoidal category ($(X,Y,e,n)$)

Objects $X,Y$ with an evaluation morphism $e:X\otimes Y\to I$ and a coevaluation morphism $n:I\to Y\otimes X$ satisfying both snake identities. These are categorical duality data, distinct from the locally convex dual pair.

The two triangular identities $(e\otimes 1_X)(1_X\otimes n)=1_X$ and $(1_Y\otimes e)(n\otimes 1_Y)=1_Y$, with canonical associators and unitors inserted, for a dual pair in a monoidal category.

The unit-valued duality map $n:I\to Y\otimes X$ of a dual pair in a monoidal category, constrained together with evaluation by both snake identities.

For a dual pair in a monoidal category, the contraction $e:X\otimes Y\to I$. In a left closed monoidal category, evaluation also denotes the adjunction map $[M,N]\otimes M\to N$.