Coherently shift-compatible functors and natural transformations

Let $k$ be a field and $A,B$ graded $k$-algebras, with ${\mathrm{GrMod}}_0(A)$, ${\mathrm{GrMod}}_0(B)$ the abelian categories of graded modules and degree-zero maps (Associative graded algebras, bimodules, and internal shifts) and with the internal-shift autoequivalences $X\mapsto X\{r\}$ of Internal shifts are autoequivalences and commute with the graded tensor product. A functor $F:{\mathrm{GrMod}}_0(A)\to {\mathrm{GrMod}}_0(B)$ (Covariant functor, identity functor, composite functor, and contravariant functor) is coherently shift-compatible when it is additive (Additive functor) and comes with a family of degree-zero $B$-linear isomorphisms ${\theta }_{X,r}:F(X\{r\})⟶F(X)\{r\},$ natural in $X$ (Natural transformation and its components, Natural isomorphism), such that for all graded modules $X$ and all $r,s\in \mathbb{Z}$ the unit and cocycle identities ${\theta }_{X,0}=1_{F(X)},{\theta }_{X,r+s}=({\theta }_{X,r}\{s\})\circ {\theta }_{X\{r\},s}$ hold under the canonical shift identifications $(X\{r\})\{s\}=X\{r+s\}$ and $F(X)\{r\}\{s\}=F(X)\{r+s\}$ provided by Internal shifts are autoequivalences and commute with the graded tensor product; here ${\theta }_{X,r}\{s\}$ is the shift of the morphism ${\theta }_{X,r}$. These are supplied equivariance data, not merely the existence of unrelated shift isomorphisms, and no particular functor is asserted to admit them.

A natural transformation $\eta :F\Rightarrow G$ between coherently shift-compatible functors is coherent, or shift-compatible, when ${\theta }_{X,r}^G{\eta }_{X\{r\}}=({\eta }_X\{r\}){\theta }_{X,r}^F$ for all $X,r$. The class of such data, with coherent transformations as morphisms, is written ${\mathrm{C}\mathrm{o}\mathrm{h}}^0(A,B)$; the sub-class used by the classification theorem consists of the $k$-linear (k-linear categories and k-linear functors) right exact coproduct-preserving members, written $\mathrm{C}\mathrm{o}\mathrm{h}\mathrm{F}\mathrm{u}\mathrm{n}(A,B)$ in Coherently shift-compatible functors and transformations form k-linear hom categories ↗. The definition assumes no commutativity of the rings beyond the central field $k$, no flatness or exactness of $F$, and uses no choice.

The coherence is a genuine restriction and not a formality: the unit and cocycle are part of the supplied data, and the equivariance square is imposed on 2-cells, so an additive functor is not coherent merely by being additive, nor automatically by being an equivalence. Hazrat's Definition 2.3.3 uses strict commutation $\phi T_{\alpha }=T_{\alpha }\phi$ with the suspension functors and, by his Remark 2.3.4, does not require natural transformations between such functors to commute with suspensions; the coherent isomorphisms and the equivariance square are the deliberate refinement used on this page, and the same 2-cells are used on both sides of the graded theorem below.