Stability conditions on morphisms in a category

Let $\mathrm{h}\mathcal{C}$ be the homotopy category of a stable infinity category $\mathcal{C}$. Then the homotopy category $\mathrm{h}{\mathcal{C}}^{{\mathrm{\Delta }}^1}$ of morphisms in the stable infinity category $\mathcal{C}$ is also triangulated. Hence the space $$\mathsf{S}\mathsf{t}\mathsf{a}\mathsf{b}$,{ $\mathrm{h}$\mathscr{C}^{\Delta^{1}}}$ofstabilityconditionson$$\mathrm{h}$\mathscr{C}^{\Delta^{1}}$ is well-defined though the non-emptiness of $\mathsf{S}\mathsf{t}\mathsf{a}\mathsf{b}\mathrm{h}{\mathcal{C}}^{{\mathrm{\Delta }}^1}$ is not obvious. Our basic motivation is a comparison of the homotopy type of $\mathsf{S}\mathsf{t}\mathsf{a}\mathsf{b}\mathrm{h}\mathcal{C}$ and that of $\mathsf{S}\mathsf{t}\mathsf{a}\mathsf{b}\mathrm{h}{\mathcal{C}}^{{\mathrm{\Delta }}^1}$. Under the motivation we show that functors $d_0$ and $d_{1} \colon \mathscr{C}^{\Delta^{1}} \rightrightarrows \mathscr{C}$inducecontinuousmapsfrom$$\mathsf{S}\mathsf{t}\mathsf{a}\mathsf{b}$ {$\mathrm{h}$\mathscr{C}}$to$$\mathsf{S}\mathsf{t}\mathsf{a}\mathsf{b}${$\mathrm{h}$\mathscr{C}^{\Delta^{1}}}$contravariantlywhere$d_{0}$(resp\mathrm{.}$d_{1}$) takes a morphism to the target (resp. source) of the morphism. As a consequence, if $\mathsf{S}\mathsf{t}\mathsf{a}\mathsf{b}\mathrm{h}\mathcal{C}$ is nonempty then so is $\mathsf{S}\mathsf{t}\mathsf{a}\mathsf{b}\mathrm{h}{\mathcal{C}}^{{\mathrm{\Delta }}^1}$. Assuming $\mathcal{C}$ is the derived infinity category of the projective line over a field, we further study basic properties of $d_0^{\ast }$ and $d_1^{\ast }$. In addition, we give an example of a derived category which does not have any stability condition.