I am trying to understand why for an additive category $\mathcal{C}$ the category $K(\mathcal{C})$ of complexes over $\mathcal{C}$ modulo homotopy is triangulated:
We take the shift functor as automorphism on $\mathcal{K(C)}$ and define distinguished triangles as those which are isomorphic to triangles $X\xrightarrow{f}Y\rightarrow C(f)\rightarrow X[1]$, where $C(f)$ is the mapping cone of $f$. See e.g. Definition 1.4.3 in Kashiwara, Shapira: "Sheaves on Manifolds" for a precise definiton.
There are detailed proves in the literature like the source mentioned above or Gelfand, Manin: "Methods of Homological Algebra". However, I would be interested in a more heuristic approach. Checking the commutativity of some diagrams with given arrows as it is done in the mentioned sources is not very hard but remains obscure to me. Does anyone have an idea how one could more naturally be led to a proof for the statement in question?