Let $C$ be any class of topological spaces, such that for any $X \in C$, if $U$ is a topological space homeomorphic to an open subset of $X$, then $U \in C$.
From now on we view $C$ as a full subcategory of the category of topological spaces.
Let $\mathrm{h}C$, the homotopy category of $C$, denote the category obtained by identifying arrows in $C$ which are parallel (same source and target) and homotopic (as continuous maps).
Let $\tau$ be the standard Grothendieck topology on $C$, generated from the Grothendieck pretopology whose covering families into $X$ are those collections $\{U_i \rightarrow X\}$ of maps which are each open embeddings and whose images cover $X$. (If I'm not mistaken, this is a Grothendieck pretopology?)
My question is, does $\tau$ "push down" to a Grothendieck topology on $\mathrm{h}C$? What would be the best way to show this?