Triangulated Structures Induced by Triangle Functors

Given a triangle functor $F\colon \A \to \B$, the authors introduce the half image $\hIm F$, which is an additive category closely related to $F$. If $F$ is full or faithful, then $\hIm F$ admits a natural triangulated structure. However, in general, one can not expect that $\hIm F$ has a natural triangulated structure. The aim of this paper is to prove that $\hIm F$ admits a natural triangulated structure if and only if $F$ satisfies the condition (SM). If this is the case, $\hIm F$ is triangle-equivalent to the Verdier quotient $\A/\Ker F$.