Stability conditions on morphisms in a category

Let $\mathrm{h}\mathscr{C}$ be the homotopy category of a stable infinity $\mathscr{C}$. Then $\mathrm{h}\mathscr{C}^{\Delta^{1}}$ morphisms in $\mathscr{C}$ is also triangulated. Hence space $\mathsf{Stab}\,{ \mathrm{h}\mathscr{C}^{\Delta^{1}}}$ stability conditions on well-defined though non-emptiness not obvious. Our basic motivation comparison type $\mathsf{Stab}{\mathrm{h}\mathscr{C}}$ and that $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$. Under we show functors $d_{0}$ $d_{1} \colon \mathscr{C}^{\Delta^{1}} \rightrightarrows \mathscr{C}$ induce continuous maps from $\mathsf{Stab} {\mathrm{h}\mathscr{C}}$ to $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$ contravariantly where (resp. $d_{1}$) takes morphism target source) morphism. As consequence, if nonempty then so Assuming derived projective line over field, further study properties $d_{0}^{*} $ $d_{1}^{*}$. In addition, give an example which does have any condition.