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  "body": "\n## Definition\n\nLet $(\\mathcal{C}, \\otimes, I, \\alpha, \\lambda, \\rho)$ and $(\\mathcal{D}, \\otimes', I', \\alpha', \\lambda', \\rho')$ be [[monoidal categories|M3MSQ3NKAE]].\nA (lax) *monoidal functor* from $\\mathcal{C}$ to $\\mathcal{D}$ is a [[functor|28NQ0KRHD3]] $F \\colon \\mathcal{C} \\to \\mathcal{D}$ together with:\n\n1. a morphism $\\varepsilon \\colon I' \\to F(I)$;\n2. a [[natural transformation|KYCPCC57FT]] $\\mu$ with components $\\mu_{A,B} \\colon F(A) \\otimes' F(B) \\to F(A \\otimes B)$;\n\nsuch that for all objects $A, B, C$ of $\\mathcal{C}$:\n\n1. associativity: $F(\\alpha_{A,B,C}) \\circ \\mu_{A \\otimes B, C} \\circ (\\mu_{A,B} \\otimes' \\id_{F(C)}) = \\mu_{A, B \\otimes C} \\circ (\\id_{F(A)} \\otimes' \\mu_{B,C}) \\circ \\alpha'_{F(A),F(B),F(C)}$;\n2. left unitality: $F(\\lambda_A) \\circ \\mu_{I,A} \\circ (\\varepsilon \\otimes' \\id_{F(A)}) = \\lambda'_{F(A)}$;\n3. right unitality: $F(\\rho_A) \\circ \\mu_{A,I} \\circ (\\id_{F(A)} \\otimes' \\varepsilon) = \\rho'_{F(A)}$.\n\nThe functor is *strong monoidal* if $\\varepsilon$ and all $\\mu_{A,B}$ are isomorphisms, and *strict monoidal* if they are identities.\n\n## Remarks\n\nExample: the forgetful functor $U \\colon (\\mathbf{Ab}, \\otimes, \\Z) \\to (\\mathbf{Set}, \\times, \\set{*})$ with $\\mu_{A,B}(a, b) = a \\otimes b$ and $\\varepsilon(*) = 1$.\nBoth sides of the associativity axiom send $((a, b), c)$ to $a \\otimes (b \\otimes c)$, and the left unit axiom reads $1 \\otimes a \\mapsto a$.\nIt is lax but not strong, since $A \\times B \\to A \\otimes B$ is not a bijection in general.\nNon-example: the forgetful functor from $\\R$-vector spaces with $\\otimes$ to $(\\mathbf{Set}, \\times)$, with $\\mu(u, v) = u \\otimes v$ but with $\\varepsilon(*) = 0$.\nThe left unit axiom fails: it would send $(*, v)$ to $0 \\otimes v = 0$, while $\\lambda'$ sends it to $v$.\nSome authors say \"monoidal functor\" for strong monoidal functors only; the lax version is then called lax monoidal.\n",
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