I asked this [question](https://math.stackexchange.com/q/2463353/660) 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](https://doi.org/10.1007/3-540-27950-4)** 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](http://www.normalesup.org/%7Eforgogozo/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:


* $F$ is a 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**,
* $F$ is a 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.



Equivalence of the definitions of a sheaf in SGA4 and in "Categories and Sheaves"
$\DeclareMathOperator\im{im}$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.


