Homotopy categories of admissible model structures on extriangulated categories

The extriangulated category is a simultaneous generalization of exact categories and triangulated categories. H. Nakaoka and Y. Palu have proved that the homotopy category of an admissible model structure on a weakly idempotent complete extriangulated category is a triangulated category. Using the classic construction of distinguished triangles given by A. Heller and D. Happel, this paper provides an alternative proof of Nakaoka - Palu Theorem. In fact, the class $\Delta$ of distinguished triangles in the present paper and the class $\widetilde{\Delta}$ of distinguished triangles in \cite{NP} have the relation $\Delta = - \widetilde{\Delta}$, and hence the two triangulated structures on the homotopy category are isomorphic.