0N8E Definition 16. Let (π,β,I,R,R~(β|β,β),R~(β,β|β))(\mathcal{C},\otimes,I,R,\tilde{R}_{(-|-,-)},\tilde{R}_{(-,-|-)}) and (πβ²,ββ²,I,Rβ²,R~(β|β,β)β²,R~(β,β|β)β²)(\mathcal{C}^{\prime},\otimes^{\prime},I,R^{\prime},\tilde{R}^{\prime}_{(-|-,-)},\tilde{R}^{\prime}_{(-,-|-)}) be braided monoidal 2-categories. A monoidal 2-functor consists of: β’ A 2-functor β±:πβπβ²\mathcal{F}:\mathcal{C}\to\mathcal{C}^{\prime} such that β±β‘(I)=Iβ²\mathcal{F}(I)=I^{\prime}. β’ A pseudonatural transformation ΞΎ:(β±βGβ±)βββ²ββββ±,\xi:(\mathcal{F}\otimes_{\rm G}\mathcal{F})\circ\otimes^{\prime}\Rightarrow\otimes\circ\mathcal{F}, β’ an invertible modification Ξ±:(1βΞΎ)βΞΎβ(ΞΎβ1)βΞΎ\alpha:(1\otimes\xi)\circ\xi\Rightarrow(\xi\otimes 1)\circ\xi, such that the following diagram commutes. β±β‘(X)ββ±β(Y)ββ±β(Z)ββ±β(W){\lx@inpgf@ignorespaces\mathcal{F}(X)\mathcal{F}(Y)\mathcal{F}(Z)\mathcal{F}(W)}β±β‘(X)ββ±β(YβZ)ββ±β(W){\lx@inpgf@ignorespaces\mathcal{F}(X)\mathcal{F}(YZ)\mathcal{F}(W)}β±β‘(XβY)ββ±β(Z)ββ±β(W){\lx@inpgf@ignorespaces\mathcal{F}(XY)\mathcal{F}(Z)\mathcal{F}(W)}β±β‘(XβYβZ)ββ±β(W){\lx@inpgf@ignorespaces\mathcal{F}(XYZ)\mathcal{F}(W)}β±β‘(X)ββ±β(Y)ββ±β(ZβW){\lx@inpgf@ignorespaces\mathcal{F}(X)\mathcal{F}(Y)\mathcal{F}(ZW)}β±β‘(X)ββ±β(YβZβW){\lx@inpgf@ignorespaces\mathcal{F}(X)\mathcal{F}(YZW)}β±β‘(XβY)ββ±β(ZβW){\lx@inpgf@ignorespaces\mathcal{F}(XY)\mathcal{F}(ZW)}β±β‘(XβYβZβW){\lx@inpgf@ignorespaces\mathcal{F}(XYZW)}2.{\lx@inpgf@ignorespaces 2.}5.{\lx@inpgf@ignorespaces 5.}3.{\lx@inpgf@ignorespaces 3.}6.{\lx@inpgf@ignorespaces 6.}1.{\lx@inpgf@ignorespaces 1.}4.{\lx@inpgf@ignorespaces 4.} 1.=^βΞΎ,ΞΎ2.=^βΞ±XβY,Z,W3.=^βΞ±X,Y,ZβW4.=^βΞ±X,YβZ,W5.=^βΞ±X,Y,Zββ±β‘(W)6.=^ββ±β(X)βΞ±Y,Z,W\begin{array}[]{llll}1.\hat{=}\>\otimes_{\xi,\xi}&2.\hat{=}\>\alpha_{X\otimes Y,Z,W}&3.\hat{=}\>\alpha_{X,Y,Z\otimes W}&4.\hat{=}\>\alpha_{X,Y\otimes Z,W}\\ 5.\hat{=}\>\alpha_{X,Y,Z}\otimes\mathcal{F}(W)&6.\hat{=}\>\mathcal{F}(X)\otimes\alpha_{Y,Z,W}&\end{array}