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 $\mathrm{\Delta }$ of distinguished triangles in the present paper and the class $\tilde{\mathrm{\Delta }}$ of distinguished triangles in \cite{NP} have the relation $\mathrm{\Delta }=-\tilde{\mathrm{\Delta }}$, and hence the two triangulated structures on the homotopy category are isomorphic.