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  "body": "\n## Definition\n\nLet $(\\mathcal{C}, \\otimes, I, \\alpha, \\lambda, \\rho)$ be a [[monoidal category|M3MSQ3NKAE]].\n$\\mathcal{C}$ is *closed* (more precisely, right closed) if for every object $B$ the functor $- \\otimes B \\colon \\mathcal{C} \\to \\mathcal{C}$ has a right [[adjoint|TZHF6ZM92S]], written $[B, -] \\colon \\mathcal{C} \\to \\mathcal{C}$.\nEquivalently, for all objects $A, B, C$ there is a bijection\n$$\\Hom_{\\mathcal{C}}(A \\otimes B, C) \\cong \\Hom_{\\mathcal{C}}(A, [B, C]),$$\nnatural in $A$ and $C$.\nThe object $[B, C]$ is called the internal hom.\n\n## Remarks\n\nIf also every functor $B \\otimes -$ has a right adjoint, $\\mathcal{C}$ is called biclosed; for a symmetric monoidal category the two notions agree.\nExample: $\\mathbf{Vect}_k$ with $[B, C]$ the space of linear maps $B \\to C$; a linear map $f \\colon A \\otimes B \\to C$ corresponds to $a \\mapsto (b \\mapsto f(a \\otimes b))$.\nNon-example: $\\mathbf{Set}$ with the disjoint union $\\sqcup$ as tensor product and the empty set as unit is monoidal but not closed.\nA left adjoint preserves the initial object, but $\\emptyset \\sqcup B = B$ is not empty when $B$ is nonempty, so $- \\sqcup B$ has no right adjoint.\nA [[Cartesian closed category|BFBNQ36QS8]] is a closed monoidal category whose tensor product is the categorical product.\n",
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