Unrestricted graded natural transformations are not determined by the regular module

For every family $({\lambda }_d)$ the components ${\eta }_X(x):={\lambda }_{d}x$ for $x\in X_d$ extend uniquely over finite homogeneous sums to degree-zero $k$-linear maps ${\eta }_X:X\to X$, and these define a natural transformation $\eta :\mathrm{i}\mathrm{d}\Rightarrow \mathrm{i}\mathrm{d}$: for a degree-zero $u:X\to Y$ and $x\in X_d$ one has $u(x)\in Y_d$, so ${\eta }_Y(u(x))={\lambda }_{d}u(x)=u({\lambda }_{d}x)=u({\eta }_X(x))$.

Such an $\eta$ is coherent for the canonical comparisons of the identity functor if and only if ${\lambda }_{d+r}={\lambda }_d$ for all $d,r$: coherence reads ${\eta }_{X\{r\}}={\eta }_X\{r\}$ as underlying maps by [L2], and on $x\in X_{e-r}=(X\{r\}{)}_e$ this is ${\lambda }_{e}x={\lambda }_{e-r}x$; taking $X=k\{e-r\}$, whose degree-$(e-r)$ piece is nonzero [L3], gives ${\lambda }_e={\lambda }_{e-r}$ for all $e,r$, while a constant family clearly satisfies the square.

Hence the coherent endomorphisms of the identity functor are exactly the constant families, and they correspond bijectively to $k$ via $\eta \mapsto {\eta }_k$, in agreement with [L1], where the coherent endomorphisms correspond to ${\mathrm{Hom}}_{k\text{-}k}(k,k)\cong k$: the constant family with scalar $\lambda$ has component $\lambda {\mathrm{i}\mathrm{d}}_X$ at every $X$, and since $T_k\cong \mathrm{i}\mathrm{d}$ by the unit isomorphism [L5] the two descriptions agree.

The family ${\lambda }_d=1$ for $d=1$ and ${\lambda }_d=0$ for $d\mathrm{\ne }1$ is nonconstant, so by step 2.1 it is not coherent, but by step 1.1 it is a nonzero natural endomorphism of the identity functor: its component at $k$ is ${\lambda }_0{\mathrm{i}\mathrm{d}}_k=0$, the same as that of the zero transformation, while its component at $k\{1\}$ is ${\lambda }_1{\mathrm{i}\mathrm{d}}_{k\{1\}}={\mathrm{i}\mathrm{d}}_{k\{1\}}\mathrm{\ne }0$ because $(k\{1\}{)}_1\mathrm{\ne }0$ [L3]; hence two distinct natural transformations of $\mathrm{i}\mathrm{d}$ have the same component at the regular module, and after transporting along $T_k\cong \mathrm{i}\mathrm{d}$ [L5] there are distinct natural transformations $T_k\Rightarrow T_k$ with the same image under $\eta \mapsto {\eta }_k$.

Collecting steps 1.1 to 3.2: the unconstrained natural endomorphisms of the identity form a family strictly larger than the coherent ones, they are not determined by their component on the regular module, and only the constant families satisfy the equivariance square; so the restriction on 2-cells in Coherently shift-compatible functors and natural transformations is genuinely needed for the classification of Graded bimodule maps classify shift-compatible transformations, exactly as Hazrat's remark on transformations between shift-commuting functors warns, and no choice is used.