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  "body": "\n## Summary\n\nA dual of an object $A$ in a [[monoidal category|M3MSQ3NKAE]] is an object $A^*$ with an evaluation $A^* \\otimes A \\to I$ and a coevaluation $I \\to A \\otimes A^*$.\nThese two maps must satisfy the zigzag identities.\nIt abstracts the relation between a finite-dimensional vector space and its [[dual space|07JBSZE0V4]].\nA dual, when it exists, is unique up to unique isomorphism.\nDuals give traces and dimensions in monoidal categories and correspond to bending strings in string diagrams.\n\n## Explanations\n\n### L1 human\n\nA finite-dimensional vector space $V$ has a dual $V^*$ of linear functionals.\nOne can pair a functional with a vector to get a number, and one can produce a canonical tensor $\\sum_i e_i \\otimes e^i$ out of nothing.\nDoing one after the other, in the right order, gives back the identity: these are the zigzag identities.\nA dual object is an object with these two maps and these two identities, in any monoidal category.\nIn pictures, the two maps are a cap and a cup, and the identities say that a zigzag in a string can be pulled straight.\n",
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