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.