Dual object

An object $A$ of a monoidal category $(\mathbf{C},\otimes,I)$ admits a dual when there exists an object $A *$ together with two morphisms

$\eta_A:I\to A^*\otimes A$ and $\varepsilon_A:A\otimes A^*\to I$

such that

$\lambda_A\circ(\varepsilon_A\otimes A)\circ\alpha_{A,A^*,A}^{-1}\circ(A\otimes\eta_A)\circ\rho_A^{-1}=\mathrm{id}_A$

and

$\rho_{A^*}\circ(A^*\otimes\varepsilon_A)\circ\alpha_{A^*,A,A^*}\circ(\eta_A\otimes A^*)\circ\lambda_{A^*}^{-1}=\mathrm{id}_{A^*}$ .

This category theory-related article is a stub. You can help Wikipedia by expanding it.