I don't understand why the definition of a sheaf (Definition 17.3.1 (ii)) given in the book
[KS] Categories and Sheaves by Kashiwara and Schapira
is equivalent to the definition of a sheaf (Definition 2.1) given in
[V] Verdier, Exposé II, SGA4, http://www.normalesup.org/~forgogozo/SGA4/02/02.pdf
To simplify, let me consider only set-valued presheaves.
Here is, in the terminology of [KS], how I understand the two definitions. (Warning: my understanding might be incorrect!)
Let $\mathcal U$ be a universe, let $X$ be a small site and let $F$ be a $\mathcal U$-set-valued presheaf over $X$. Then:
$\bullet\ F$ is sheaf in the sense of [V] if $F(f)$ is an isomorphism for any $A$ in $(\mathcal C_X)^\wedge$, any $U$ in $\mathcal C_X$, and any local isomorphism $f:A\to U$ which is a monomorphism,
$\bullet\ F$ is sheaf in the sense of [KS] if $F(f)$ is an isomorphism for any $A$ in $(\mathcal C_X)^\wedge$, any $U$ in $\mathcal C_X$, and any local isomorphism $f:A\to U$.
(The difference is that the local isomorphism $f$ is supposed to be a monomorphism in Verdier's definition.)
A "[KS]-sheaf" is of course a "Verdier-sheaf", but I'm unable to prove the converse.