Equivalence of the definitions of a sheaf in SGA4 and in "Categories and Sheaves" I asked this question on Mathematics Stack Exchange, but got no answer.
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 "V-sheaf", but I'm unable to prove the converse.
 A: Actually I couldn't quite figure out how to do a Ken Brown sort of argument, but here's an argument that works:
Let $L$ denote the usual sheafification functor, a la Grothendieck and Verdier etc. Then I claim it's enough to show $L$ sends local epimorphisms to epimorphisms and local monomorphisms to monomorphisms. Indeed, if this is the case then $L$ takes local isomorphisms to epi-monomorphisms, and such things are isomorphisms in toposes. 
Since $L$ preserves finite limits, we need only check that $L$ takes local epis to epis (since $A \to B$ is a local mono iff $A \to A\times_BA$ is a local epi). But that's not so bad: If $A \to B$ is a local epimorphism then the map $im(A \to B) \to B$ is a local epimorphism and a monomorphism, and hence a monic, local isomorphism. But $L$ preserves images (since they're computed as colimits and L preserves those) and takes monic, local isomorphisms to isomorphisms. Thus $LA \to LB$ has the property that $im(LA \to LB) \to LB$ is an isomorphism (NB: that image is computed in the category of sheaves), and so the map is epi in the category of sheaves, which is what we wanted.
