Let $\mathcal{C}$ be some category. One way to map this category into a triangulated category is to take the associated simplicial category $s\mathcal{C}$ (which is an $\infty$-category) and take the homotopy category $\text{Ho}(s\mathcal{C})$ of the simplicial category (which is triangulated since it is the homotopy category of an $\infty$-category). Then we get a natural map
$$\mathcal{C}\rightarrow \text{Ho}(s\mathcal{C}).$$
An example of this construction would be to set $\mathcal{C}=A$ for some abelian category, in which case by Dolb-Kan, we have $\text{Ho}(s\mathcal{C})\cong D(A).$ My question is:

Does $\text{Ho}(s\mathcal{C})$ satisfy some universal property?

i.e. is it the "universal triangulated category" associated to $\mathcal{C}$ in some sense, i.e. if $\mathcal{T}$ is a triangulated category and $\mathcal{C}\rightarrow \mathcal{T}$ satisfying some properties, does this factor through $\mathcal{C}\rightarrow \text{Ho}(s\mathcal{C})?$