A normalized group 4-cocycle gives a fusion 2-category 2Vect_G^omega

For homogeneous inputs of grades $g,h,k,l,m$, the two composites of pentagonators have scalar coefficients $\omega (h,k,l,m)\omega (g,hk,l,m)\omega (g,h,k,lm)\text{and}\omega (gh,k,l,m)\omega (g,h,kl,m)\omega (g,h,k,l)\mathrm{.}$ They agree by the 4-cocycle equation in [F1]. On arbitrary objects both pastings are direct sums of these homogeneous components, and their naturality makes the componentwise equality an equality of modifications. Every coherence equation with a unit input reduces to the identity scalar by normalization in [F1].

Let $\mathcal{A},\mathcal{B}$ have respectively $n_g,m_g$ simple labels in grade $g$. A grading-preserving linear functor $\mathcal{A}\to \mathcal{B}$ is a family of matrices of finite-dimensional vector spaces, one $m_g$-by-$n_g$ matrix for each $g$. Consequently its hom category is a finite product of finite matrix categories over ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}$. Each matrix category is semisimple: its simple objects have one copy of $\mathbb{C}$ in a single matrix entry and zero elsewhere, and every matrix is a finite direct sum of these. Composition is matrix multiplication with direct sums and tensor products of finite-dimensional vector spaces. Object direct sums are concatenation of the finite simple-label sets. This proves the hom-category and direct-sum clauses of [F3].

If $F:\mathcal{A}\to \mathcal{B}$ is represented by matrices $V_{ji}$, define its right adjoint by the transposed matrices $(F^R{)}_{ij}=V_{ji}^{\ast }$, where $V^{\ast }$ is the ordinary finite-dimensional vector-space dual; define its left adjoint using the opposite duality. The unit and counit are the direct sums of the vector-space coevaluation and evaluation maps. Each triangle composite is the identity on every matrix entry by the vector-space dual-basis identities, so these are left and right adjoints.

Let $S={⨆}_{g\in G}{S}_g$ be the finite simple-label set of $\mathcal{A}\simeq {\mathrm{V}\mathrm{e}\mathrm{c}}^S$. A grading-preserving monad $T$ has finite matrix entries $T_{t,s}$, zero unless $s,t$ have the same grade. Its unit and multiplication satisfy the monad unit and associativity laws; these are exactly the identity and composition laws for the finite $k$-linear category with objects $S$ and morphism spaces $T_{t,s}$. Its finite-dimensional category algebra $R_T={⨁}_{s,t\in S}{T}_{t,s}$ has unit the sum of the object identities, and a $T$-algebra is exactly a family $(M_s{)}_{s\in S}$ with compatible left $R_T$-action maps $T_{t,s}\otimes M_s\to M_t$. The separability section $\sigma :T\Rightarrow T^2$ gives $E=\sigma \eta (1)={\sum }_i{x}_i\otimes y_i\in R_T\otimes R_T$; the equations $\mu \sigma =1_T$ and $T$-bilinearity of $\sigma$ say respectively ${\sum }_i{x}_i{y}_i=1_{R_T}$ and $aE=Ea$ for every $a\in R_T$, component by component. For a surjective homomorphism $p:M\to N$ of finite-dimensional left $R_T$-modules, choose a $k$-linear section $s$ and set $\bar{s}(n)={\sum }_i{x}_{i}s(y_{i}n)$. Then $p\bar{s}(n)={\sum }_i{x}_i{y}_{i}n=n$, while $\bar{s}(an)={\sum }_i{x}_{i}s(y_{i}an)={\sum }_i(a{x}_i)s(y_{i}n)=a\bar{s}(n)$ by $Ea=aE$. Thus every module epimorphism splits; every finite-dimensional module is a quotient of a finite direct sum of copies of the regular module, by choosing a finite vector-space generating set, so it is projective and the finite-dimensional module category is semisimple. Every simple module is a quotient of the regular module, which has finite length, so there are finitely many simple classes. Since $R_T$ has no entries between different grades, its central grade idempotents split this Eilenberg–Moore category into finitely many finite semisimple grade pieces. The free functor $F$ sends $X$ to $(TX,{\mu }_X)$ and is left adjoint to the forgetful functor $U$: the adjunction bijection sends an algebra map $f:TX\to M$ to $f{\eta }_X$, with inverse $g\mapsto a_M\circ T(g)$ for a $T$-algebra action $a_M$. The monad and algebra laws verify that these maps are inverse. Both functors preserve grades, and $UF=T$ with unit $\eta$ and multiplication $\mu$. Hence the Eilenberg–Moore category and the adjunction $F⊣U$ give the required separable-monad splitting in $2{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_G^{\omega }$.

For completeness the splitting has the required section of its counit. For a $T$-algebra $(x,a)$ define $s_{(x,a)}=T(a){\sigma }_x{\eta }_x:x\to Tx$, as a map to the free algebra $(Tx,{\mu }_x)$. The algebra law and $\mu \sigma =1$ give $as=aT(a){\sigma }_x{\eta }_x=a{\mu }_x{\sigma }_x{\eta }_x=a{\eta }_x=1_x$. To check that $s$ is an algebra map, naturality and the two $T$-bimodule equations for $\sigma$ give $\begin{array}{rl}{\displaystyle sa}& {\displaystyle =T(a){\sigma }_x{\eta }_{x}a=T(a){\sigma }_{x}T(a){\eta }_{Tx}=T(a)T^2(a){\sigma }_{Tx}{\eta }_{Tx}}\\ {\displaystyle }& {\displaystyle =T(a)T({\mu }_x){\sigma }_{Tx}{\eta }_{Tx}=T(a){\sigma }_x{\mu }_x{\eta }_{Tx}=T(a){\sigma }_x,}\\ {\displaystyle {\mu }_{x}T(s)}& {\displaystyle ={\mu }_x{T}^2(a)T({\sigma }_x)T({\eta }_x)=T(a){\mu }_{Tx}T({\sigma }_x)T({\eta }_x)}\\ {\displaystyle }& {\displaystyle =T(a){\sigma }_x{\mu }_{x}T({\eta }_x)=T(a){\sigma }_x\mathrm{.}}\end{array}$ Thus $sa={\mu }_{x}T(s)$. For an algebra map $h:(x,a)\to (y,b)$, the identity $ha=bT(h)$ and naturality of $\sigma ,\eta$ imply $T(h)s_{(x,a)}=s_{(y,b)}h$. Every component preserves grades, since $T,a,\sigma ,\eta$ do. Hence $s:1\Rightarrow FU$ is a grading-preserving natural section of the counit $FU\Rightarrow 1$, an actual $2$-morphism in the Eilenberg–Moore endomorphism hom category required by [F3].

Tensoring linear functors on separate factors has the canonical comparison that sends an elementary tensor $x\otimes y$ to $F(x)\otimes G(y)$. The two sides of each interchange equation therefore agree on elementary tensors and hence on the entire tensor product. The scalar in [F1] commutes with every linear map, so the pentagonator is natural in 1- and 2-morphisms. Thus the prescribed operations satisfy the pseudofunctor and naturality parts of the weak monoidal 2-category definition.

The identity functor of a finite semisimple category with simple labels $s_1,\dots ,s_n$ is the direct sum of the $n$ projection-inclusion functors for those labels. Each summand is simple, so the identity is simple exactly when $n=1$. The only simple objects of $2{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_G^{\omega }$ are therefore the one-label categories ${\delta }_g$, one for each $g\in G$. In particular ${\delta }_e$ is the unique simple summand of the monoidal unit and is simple.

For an object with simple labels $s_{g,i}$, put the dual label $s_{g,i}^{\mathrm{\#}}$ in grade $g^{-1}$. Evaluation sends $(s_{g,i},s_{h,j}^{\mathrm{\#}})$ to $\mathbb{C}$ when $h=g$ and $j=i$, and to $0$ otherwise; coevaluation sends $\mathbb{C}$ to the direct sum of $s_{g,i}^{\mathrm{\#}}\otimes s_{g,i}$ over all labels. Each right-dual triangle composite is the identity on every simple label, since only its matching diagonal summand survives; reversing evaluation and coevaluation gives the same check for left duals. Thus both cusp transformations have scalar $1$. These composites use only the associator and unitors on at most three object factors; the 4-cocycle enters the four-input pentagonator already checked in step 1.1, and normalized unit cases contribute scalar $1$. This supplies duals without asserting a pivotal or spherical enhancement.