Remove the identity assumption in the Γ-induced isomorphism on reduced K-theory

Establish whether, for an abelian category C with endofunctors G and H such that G is right exact and H is left exact, the homomorphism on reduced K-theory induced by the composition functor Γ: End(C; G, H; n,1) → End(C; G, H; 1,n)—namely, the map in equation (eq:uncompose_2)—is an isomorphism without requiring the assumption that either G = 1_C or H = 1_C. Resolving this would determine whether the isomorphism of Corollary (lem:multiple_uncompose) extends to the fully bitwisted setting.

Background

The paper defines generalized endomorphism categories End(C; G, H; n,1) and End(C; G, H; 1,n), where objects are tuples of maps (f_j: G(c_{j+1}) → H(c_j)) with c_0 = c_n, and introduces a composition functor Γ that composes these maps to produce a single map G^n(c) → H^n(c).

They prove in Corollary (lem:multiple_uncompose) that the induced homomorphism on reduced K_0,  ilde{K}_0(C; G, H; n,1) →  ilde{K}_0(C; G, H; 1,n), is an isomorphism when either G or H is the identity functor. The authors note that they suspect this assumption is unnecessary—which would allow treating bitwisted endomorphisms without this restriction—but they cannot currently prove the general case.

Clarifying whether Γ induces an isomorphism on reduced K-theory for arbitrary right-exact G and left-exact H would strengthen the structural results and potentially remove technical assumptions in later sections (e.g., for constructing Verschiebung without requiring G = 1_C or H = 1_C).