$$0 \to A \xrightarrow{f}  B \xrightarrow{g} C \to 0$$

which is called an almost split sequence if
1) It does not split 
2) $f$ is a left almost minimal split and $g$ is a right almost minimal split (a lot is here packed, please see the definitions in AR paper, III)
3) $A$ and $C$ are indecomposable in $rep_k Q$ (category of finite-dimensional representations of quiver Q)

Such AR sequences are the main interest of Auslander-Reiten theory.  There is an AR sequence that can be easily observed from the AR quiver, which is

$$ 0 \to \tau V \to E \to V \to 0$$

where $V$ is some indecomposable and non-projective and $\tau V$ is the AR translate of $V$, which is the main subject of this post. There does not exist an AR translation of a projective representation since we begin with a projective resolution of an indecomposable, and if that is a projective, then it is not very interesting, and we fail to get a minimal projective resolution.

A projective resolution of a representation is helpful because of the equivalence between projectives and injectives using the Nakayama functor (which we may discuss in another post, but you may see this). The general idea is that  $$\nu : proj_k Q \to inj_k Q$$ and similarly, there exists an inverse $\nu^{-1}$. So, for any indecomposable $V$, we write its projective resolution as $$\cdots P_2 \to P_1 \to P_0 \to V \to 0$$ where $P_0 \to V$ is a surjection given that our category has enough projectives and, of course, our category is hereditary. We then have the AR translation of $V$ given by (for a minimal projective resolution)

$$0 \to \tau V  \to \nu P_1 \to \nu P_0$$  where $\nu$ is the Nakayama functor $\nu = DHom_A(-,A)$ which sends a projective to its corresponding injective.

For example, in $A_3$, if we take $V=110$ (where the digits are the dimension vectors of the vector spaces of representation $k \to k \to 0$), then we have the minimal projective resolution as $$ 001 \to 111 \to 110\to 0$$ and the AR translation of $110$ is $$0 \to \tau V = 011 \to 111 \to 100$$ (where $\tau V = ker (\nu (P_1) \to \nu(P_0))$ which, if you observe in the AR quiver, then it is the indecomposable which you reach after one step moving to the left side (if you can). Similarly, we can find the AR translation of all the other non-projective indecomposable. Using $\tau V$, we define the AR sequence $0 \to \tau V \to E \to V \to 0$ which is an almost split sequence (and dually, $0 \to V \to E \to \tau^{-1} V\to 0$). From here, we land into the discussion of extensions (derived functors!) and how to define these extensions using the AR quiver for which Auslander-Reiten wrote their beautiful formula (some other time).

If you could tell, there is a lot of homological algebra (and category theory) that exists between these steps, and Auslander-Reiten theory has interesting consequences in triangulated settings as well (see Happel). You can also refer to the notes here.

A triangulated category is an additive category with a shift functor that satisfies some axioms (see below). It is represented by $(\mathcal{T},[1],\Delta)$ where $[1]$ is a shift-functor (an autoequivalence $[1]: \mathcal{T} \to \mathcal{T}$) and $\Delta$ are distinguished triangles. A distinguished triangle is of the form

$$X \to Y \to Z \to X[1]$$

and the triangulated category $\mathcal{T}$ satisfies Verdier's axioms (of which an important one is the octahedral axiom, TR4). A good example of a triangulated category is the derived category of an abelian category. Anyhow, I will stop this definition by mentioning that we can see these distinguished triangles as a replacement of short exact sequences in abelian categories. For more on these triangulated categories, please see.

Our goal here is to see why there is no canonical functorial definition of (co)limits in a triangulated category. When working with some nice definition like $Qcoh(X_i) = A_i-mod$ where we have taken an open cover on $X$, we want the gluing to be defined. This motivates one to develop dg-enhancements, stable $\infty$-category, and so on (for some other day) to recover 'functorial cones'. However, I must mention that there exists a notion of homotopy (co)limits (see this work). In a distinguished triangle diagram, those come at the third vertex ($Z$). I do not know much about this theory, but they face certain restrictions too (like totalization, non-functoriality, and so on). Anyway, let us proceed with our problem. We will be mostly dealing with Stevenson's argument.

The argument is as follows. If an idempotent complete triangulated category $\mathcal{T}$ which  admits a functorial cone, then it is a semisimple abelian category. A functorial cone is a functor defined as $\text{cone}: Mor(\mathcal{T}) \to \mathcal{T}$ where $Mor(\mathcal{T})$ is a category with objects $f: X \to Y$ for $X, Y$ in $\mathcal{T}$ (also called arrow category). For any morphism $f$, we can write a distinguished triangle

and this triangle satisfies Verdier's axioms. Any morphism between two morphisms $f, f'$ is a commutative square. And there exists a coherent choice, which means that two cones of the same morphism are equivalent up to canonical isomorphism. For a non-functorial cone, cones are defined up to non-unique isomorphisms. Hence, there does not exist a coherent choice for a cone that is natural for the commutative square of morphisms.

Given a functorial cone in a category, this implies the existence of functorial weak colimits, and if a category also admits split idempotents, there exist actual colimits. Let us say our $\mathcal{T}$ is such a category. The functorial cone provides a functorial weak cokernel. In $\mathcal{T}$ for a functorial cone, there would exist an actual cokernel. Similarly, the category has an actual kernel as well. This makes $\mathcal{T}$ into an abelian category (it was already defined to be an additive category). Now, in the triangulated category $\mathcal{T}$, all monomorphisms and epimorphisms split, which makes $\mathcal{T}$ into a semi-simple abelian category. This is proposition 3.1, Stevenson. Without idempotents, one would still arrive at the same conclusion.

The only problem is that there are only trivial examples of triangulated categories that are semi-simple abelian categories. Hence, non-trivial triangulated categories cannot admit functorial cones. For example, the derived category of bounded coherent sheaves on a variety, the stable homotopy category of spectra, and so on are some examples that are not semi-simple categories. This brings us to conclude that a triangulated category admits non-functorial cones. To restore functoriality, one would need more than a triangulated structure, and such motivations will be discussed in due course.