Left dual and right dual object

Let $(\mathcal{C},\otimes ,\mathbf{1},\alpha ,\lambda ,\rho )$ be a monoidal category and let $X$ be an object of $\mathcal{C}$.

A left dual of $X$ is an object $X^{\vee }$ together with morphisms

${\mathrm{ev}}_X:X^{\vee }\otimes X\to \mathbf{1},{\mathrm{coev}}_X:\mathbf{1}\to X\otimes X^{\vee }$

such that the composites

$X\stackrel{{\lambda }_X^{-1}}{\to }\mathbf{1}\otimes X\stackrel{{\mathrm{coev}}_X\otimes 1_X}{\to }(X\otimes X^{\vee })\otimes X\stackrel{{\alpha }_{X,X^{\vee },X}}{\to }X\otimes (X^{\vee }\otimes X)\stackrel{1_X\otimes {\mathrm{ev}}_X}{\to }X\otimes \mathbf{1}\stackrel{{\rho }_X}{\to }X$

and

$X^{\vee }\stackrel{{\rho }_{X^{\vee }}^{-1}}{\to }{X}^{\vee }\otimes \mathbf{1}\stackrel{1_{X^{\vee }}\otimes {\mathrm{coev}}_X}{\to }{X}^{\vee }\otimes (X\otimes X^{\vee })\stackrel{{\alpha }_{X^{\vee },X,X^{\vee }}^{-1}}{\to }(X^{\vee }\otimes X)\otimes X^{\vee }\stackrel{{\mathrm{ev}}_X\otimes 1_{X^{\vee }}}{\to }\mathbf{1}\otimes X^{\vee }\stackrel{{\lambda }_{X^{\vee }}}{\to }{X}^{\vee }$

are identity morphisms.

A right dual of $X$ is an object ${}^{\vee }X$ together with morphisms

${\mathrm{ev}}_X^{\mathrm{\prime }}:X\otimes {}^{\vee }X\to \mathbf{1},{\mathrm{coev}}_X^{\mathrm{\prime }}:\mathbf{1}\to {}^{\vee }X\otimes X$

such that the mirror composites

$X\stackrel{{\rho }_X^{-1}}{\to }X\otimes \mathbf{1}\stackrel{1_X\otimes {\mathrm{coev}}_X^{\mathrm{\prime }}}{\to }X\otimes ({}^{\vee }X\otimes X)\stackrel{{\alpha }_{X,{}^{\vee }X,X}^{-1}}{\to }(X\otimes {}^{\vee }X)\otimes X\stackrel{{\mathrm{ev}}_X^{\mathrm{\prime }}\otimes 1_X}{\to }\mathbf{1}\otimes X\stackrel{{\lambda }_X}{\to }X$

and

${}^{\vee }X\stackrel{{\lambda }_{{}^{\vee }X}^{-1}}{\to }\mathbf{1}\otimes {}^{\vee }X\stackrel{{\mathrm{coev}}_X^{\mathrm{\prime }}\otimes 1_{{}^{\vee }X}}{\to }({}^{\vee }X\otimes X)\otimes {}^{\vee }X\stackrel{{\alpha }_{{}^{\vee }X,X,{}^{\vee }X}}{\to }{}^{\vee }X\otimes (X\otimes {}^{\vee }X)\stackrel{1_{{}^{\vee }X}\otimes {\mathrm{ev}}_X^{\mathrm{\prime }}}{\to }{}^{\vee }X\otimes \mathbf{1}\stackrel{{\rho }_{{}^{\vee }X}}{\to }{}^{\vee }X$

are identity morphisms.

This page uses EGNO's convention: the word "left" refers to the side on which the dual object sits in the evaluation map.