Etale spaces of a presfeaf and the associated sheaf

Etale spaces of a presfeaf and the associated sheaf

Given a presheaf $\mathcal{F}$on a topological space $X$, one can
construct the etale space $\pi_1 : Y_1\to X$. Let us now look at the
associated sheaf $\mathcal{F}^+$ as a presheaf and construct the etale
space $\pi_2 :Y_2\to X$ corresponding to the presheaf $\mathcal{F}^+$. My
question is : what is the relation of $Y_1$ to $Y_2$ ? There is a natural
map $\tau : \mathcal{F} \to \mathcal{F}^+ $ which gives rise to maps at
the stalk level (by taking direct limits) and hence a map $\tau_{ES} : Y_1
\to Y_2 $. Is this map a homeomorphism ? (If $\mathcal{F}$ were a sheaf to
begin with, $\tau$ would have been an isomorphism and hence $\tau_{ES}$
would have been a homeomorphism. However I am not sure if this is the case
even if we begin with a presheaf which is not a sheaf.)