The mapping cone construction does not give the category of chain complexes the structure of a triangulated category. While it does allow us to complete every chain map $f:A\to B$ to a cone, these set of exact triangles do not satisfy the axiom TR1. For example, the mapping cone of the identity $\mathrm{Cone}(\mathrm{id}:A\to A)$ is usually not isomorphic to $0$ in the category of chain complexes. This problem can be solved by looking instead at the homotopy category of chain complexes.