A graded algebra isomorphism transports the graded module category, its Grothendieck groups and its dualities

Restriction of scalars is an equivalence. Pulling back a graded left $H$-module $M$ along $\rho$ gives a graded left $R$-module with the same underlying graded vector space and action $r\cdot m=\rho (r)m$, and pulling back along ${\rho }^{-1}$ is a functor in the other direction; the two composites are the identity on underlying graded vector spaces and hence on objects and morphisms [F1]. Because $\rho$ is a bijection and preserves degree, the functor is exact, preserves finite-dimensionality, and satisfies ${\rho }^{\ast }M\{r\}=({\rho }^{\ast }M)\{r\}$ [F1].

Projectives, Hom and composition multiplicities. A finitely generated graded projective is a direct summand of a finite free graded module; pulling back preserves finite free modules and retracts, so it preserves finitely generated graded projectives [F1]. It is the identity on graded Hom spaces ${\mathrm{Hom}}_{R,d}(P,{\rho }^{\ast }M)={\mathrm{Hom}}_{H,d}({\rho }_{\ast }P,M)$ and on shifts, so it preserves graded Hom dimensions and consequently composition multiplicities of graded simples together with their shifts; the graded Hom pairing of [F2] is therefore identified [F2].

The Grothendieck identifications. Since both inverse functors are exact and preserve split exact sequences, finite dimension and the shift, restriction along ${\rho }^{-1}$ induces $A$-linear isomorphisms $K_0^{\mathrm{g}\mathrm{r}}(R)\to K_0^{\mathrm{g}\mathrm{r}}(H)$ and $G_0^{\mathrm{g}\mathrm{r}}(R)\to G_0^{\mathrm{g}\mathrm{r}}(H)$; the underlying module of a projective is carried to the underlying module, so the Cartan homomorphisms and the forgetful maps $\zeta$ are intertwined, and $q$ is preserved because shifts are [F1, F2].

Dualities. Suppose $\rho$ intertwines the stars. For a finite-dimensional graded module $M$, the transported duality is $(a\cdot f)(m)=f(a^{{\ast }_H}m)$ on the dual space; substituting $a=\rho (x)$ and using $\rho (x^{{\ast }_R})=\rho (x{)}^{{\ast }_H}$ shows that the duality computed with ${\ast }_R$ on ${\rho }^{\ast }M$ equals the duality computed with ${\ast }_H$ on $M$, so the two diagram dualities $D_{\psi }$ are intertwined [F3]. The same substitution on $\mathrm{HOM}(P,R)\cong \mathrm{HOM}({\rho }_{\ast }P,H)$ identifies the projective Hom-dualities $\mathrm{\#}$; consequently normalized self-dual simples and normalized self-dual indecomposable projectives are preserved, since normalization is a shift-fixing condition [F3].

The application. For each nonzero content sector the block isomorphism ${\rho }_{\alpha }$ is a graded algebra isomorphism [F3]. Defining ${\ast }_H={\rho }_{\alpha }{\ast }_R{\rho }_{\alpha }^{-1}$ gives an anti-automorphism because a conjugation of an anti-automorphism is again one, it is an involution because ${\ast }_R$ is, it preserves the grading because ${\rho }_{\alpha }$ does, and it fixes the named images ${\rho }_{\alpha }(e(i)),{\rho }_{\alpha }(y_r),{\rho }_{\alpha }({\psi }_r)$ whenever ${\ast }_R$ fixes the corresponding KLR generators; the intertwining ${\rho }_{\alpha }{\ast }_R={\ast }_H{\rho }_{\alpha }$ holds by construction, so the duality conclusion of step 2.2 applies blockwise [F3]. No identification of ${\ast }_H$ with the raw cellular anti-involution is made, and no product-fixedness is inferred from fixedness of factors.

No choice. The construction is the inverse-pair of functors induced by an explicit bijection, and every verification is a one-line substitution on generators or Hom spaces; no family of nonempty sets is selected. Hence no choice is used.