In the paper Some new axioms for triangulated categories, Neeman introduces a list of axioms on an additive category $\mathbf T$ with a given self-equivalence $\Sigma\colon \mathbf T\to \mathbf T$. These axioms are stronger than that of being triangulated (in the sense that the structure induced by these axioms canonically implies the existence of a triangulated structure) and allow to make some choices more canonical (which is extremely useful as many constructions in triangulated categories are only unique up to non-unique isomorphism).
In particular, to have a Neeman's triangulated structure on $\mathbf T$, one asks that there exists a category $\frak S$ of triangles with a functor $F\colon \mathfrak S\to \mathrm{CTria}(\mathbf T)$ to the category of candidate triangles in $\mathbf T$, and that this functor satisfies a long list of $10$ axioms.
The first of these axioms is the following
(GTR.1) Given a candidate triangle $$T:x\overset{\alpha}{\to} y\overset{\beta}{\to} z\overset{\gamma}{\to} \Sigma x$$ the set of triangles $S$ in $\frak S$ such that $F(S)= T$ is a principal homogeneous space under the action of the Abelian group $$\overline{E(T)}:=\frac{\{\phi\in\mathbf T(z,z):\gamma\phi=0,\phi\beta=0\}}{\{\beta\theta\gamma:\theta\in\mathbf T(\Sigma x,y)\}}$$
Then, the axiom (GTR.3) asks that the action defined in (GTR.1) is compatible with morphisms in the obvious way.
Maybe I am missing something, but the way in which the action should be compatible with morphisms is not obvious to me. I know this can be tedious, but can anyone re-state this axiom (GTR.3) in a very explicit form?