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  "body": "\n## Definition\n\nLet $\\mathcal{B}$ and $\\mathcal{B}'$ be [[bicategories|TAHV24ED74]] with associators $\\alpha, \\alpha'$ and unitors $\\lambda, \\rho, \\lambda', \\rho'$.\nA *pseudofunctor* $F \\colon \\mathcal{B} \\to \\mathcal{B}'$ consists of:\n\n1. for each object $a$ of $\\mathcal{B}$, an object $F(a)$ of $\\mathcal{B}'$;\n2. for each pair $a, b$, a [[functor|28NQ0KRHD3]] $F_{a,b} \\colon \\mathcal{B}(a, b) \\to \\mathcal{B}'(F(a), F(b))$;\n3. for composable $f \\colon a \\to b$ and $g \\colon b \\to c$, invertible 2-morphisms $\\phi_{g,f} \\colon F(g) \\circ F(f) \\Rightarrow F(g \\circ f)$, [[natural|KYCPCC57FT]] in $f$ and $g$;\n4. for each object $a$, an invertible 2-morphism $\\phi_a \\colon 1_{F(a)} \\Rightarrow F(1_a)$.\n\nThese data satisfy, for all $f \\colon a \\to b$, $g \\colon b \\to c$, $h \\colon c \\to d$:\n\n1. $F(\\alpha_{h,g,f}) \\cdot \\phi_{h \\circ g, f} \\cdot (\\phi_{h,g} \\circ 1_{F(f)}) = \\phi_{h, g \\circ f} \\cdot (1_{F(h)} \\circ \\phi_{g,f}) \\cdot \\alpha'_{F(h),F(g),F(f)}$;\n2. $F(\\lambda_f) \\cdot \\phi_{1_b, f} \\cdot (\\phi_b \\circ 1_{F(f)}) = \\lambda'_{F(f)}$;\n3. $F(\\rho_f) \\cdot \\phi_{f, 1_a} \\cdot (1_{F(f)} \\circ \\phi_a) = \\rho'_{F(f)}$.\n\nHere $\\cdot$ is vertical composition of 2-morphisms.\n\n## Remarks\n\nIf the $\\phi_{g,f}$ and $\\phi_a$ are only required to be 2-morphisms, not invertible, $F$ is called a lax functor; if they are identities, $F$ is a strict 2-functor.\nExample: regard the category of rings as a [[2-category|CQACXG10JZ]] with only identity 2-morphisms.\nSend a ring $R$ to its category of left modules, and a ring map $u \\colon R \\to S$ to extension of scalars $S \\otimes_R -$.\nThe canonical isomorphisms $T \\otimes_S (S \\otimes_R M) \\cong T \\otimes_R M$ and $R \\otimes_R M \\cong M$ give $\\phi$, and the axioms hold because these isomorphisms are the standard ones on elementary tensors.\nNon-example: a monoidal category is a one-object bicategory, and a lax [[monoidal functor|PY430TXMG0]] gives a lax functor.\nThe forgetful functor $(\\mathbf{Ab}, \\otimes) \\to (\\mathbf{Set}, \\times)$ with $\\phi(a, b) = a \\otimes b$ is a lax functor but not a pseudofunctor, since $A \\times B \\to A \\otimes B$ is not invertible.\n",
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