On the nLab, given a local $S$-topos $E$, a concrete sheaf is defined as an object that is separated with respect to the local isomorphisms (the morphisms that are inverted by the global sections functor $\Gamma:E\rightarrow S$): https://ncatlab.org/nlab/show/concrete+sheaf#in_a_local_topos

However, separated means that these morphisms are merely send to monos and not to isos. So I cannot see why a concrete sheaf would, in particular, be a sheaf.

I'm pretty sure I made a mistake in my reasoning, so I was wondering if somebody had a simple explanation.