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  "body": "\n## Definition\n\nLet $(\\mathcal{C}, \\otimes, I, \\alpha, \\lambda, \\rho)$ be a [[monoidal category|M3MSQ3NKAE]] and let $A$ be an object of $\\mathcal{C}$.\nA *dual* of $A$ is an object $A^*$ together with morphisms $\\mathrm{ev} \\colon A^* \\otimes A \\to I$ and $\\mathrm{coev} \\colon I \\to A \\otimes A^*$ such that:\n\n1. $\\rho_A \\circ (\\id_A \\otimes \\mathrm{ev}) \\circ \\alpha_{A,A^*,A} \\circ (\\mathrm{coev} \\otimes \\id_A) \\circ \\lambda_A^{-1} = \\id_A$;\n2. $\\lambda_{A^*} \\circ (\\mathrm{ev} \\otimes \\id_{A^*}) \\circ \\alpha^{-1}_{A^*,A,A^*} \\circ (\\id_{A^*} \\otimes \\mathrm{coev}) \\circ \\rho_{A^*}^{-1} = \\id_{A^*}$.\n\nThese are the zigzag (or triangle) identities.\nIn this situation $A^*$ is called a right dual of $A$, and $A$ a left dual of $A^*$.\nSome authors exchange the words left and right.\n\n## Remarks\n\nExample: in $\\mathbf{Vect}_k$, a finite-dimensional space $V$ with basis $e_1, \\ldots, e_n$ has the [[dual space|07JBSZE0V4]] $V^*$ with dual basis $e^1, \\ldots, e^n$ as dual.\nTake $\\mathrm{ev}(f \\otimes v) = f(v)$ and $\\mathrm{coev}(1) = \\sum_i e_i \\otimes e^i$.\nThe first identity sends $v$ to $\\sum_i e_i \\, e^i(v) = v$, and the second sends $f$ to $\\sum_i f(e_i)\\, e^i = f$.\nNon-example: an infinite-dimensional space $V$ has no dual in $\\mathbf{Vect}_k$.\nThe element $\\mathrm{coev}(1)$ is a finite sum $\\sum_{i=1}^m v_i \\otimes w_i$, so the first identity would express every $v \\in V$ as a combination of $v_1, \\ldots, v_m$, which is impossible.\nDuals are unique up to unique isomorphism compatible with $\\mathrm{ev}$ and $\\mathrm{coev}$.\nIn a symmetric monoidal category left and right duals coincide, and the trace of $f \\colon A \\to A$ is $\\mathrm{ev} \\circ \\gamma_{A,A^*} \\circ (f \\otimes \\id_{A^*}) \\circ \\mathrm{coev} \\colon I \\to I$.\n",
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