I'd like to prove that two definitions of sheaves on a site are equivalent, but I'm having trouble proving one direction.


Let $C$ be a category with pullbacks. Let $(C,T)$ be a site defined through a collection $\Phi$ of covering families. (So that a subfunctor of $Hom(-,U)$ is a covering sieve in $T$ if and only if it contains a covering family in $\Phi$).


Let $F$ be a presheaf on $C$, taking values in some complete category.


Then it should be the case that the following are equivalent characterisations of $F$ being a sheaf.


1) For each $U\in C$, and each covering sieve $R \subset Hom(-,U)$, the map $F(U) \to {{\lim \atop \leftarrow} \atop {V\to U \in R}} F(V)$ is an isomorphism.


2) The following diagram is an equalizer for each $U$ in $C$, and covering family {$U\_\alpha \to U$} in $\Phi$:
$$ F(U) \to \prod\_\gamma F(U\_\gamma) {\rightarrow \atop \rightarrow} \prod\_{(\alpha,\beta)} F(U\_\alpha \times\_U U\_\beta)$$


**Question:** How do you prove $2) \implies 1)$?. (I've proved $1) \implies 2)$)


Here's my proof attempt of $2) \implies 1)$ so far: 


$\bullet$ Suppose we have a covering family {$U\_\alpha \to U$} in $\Phi$,and that it generates a sieve $S$. I proved that the following is an equalizer:


$$ {{\lim \atop \leftarrow} \atop {V\to U \in S}} F(V) \to \prod\_\gamma F(U\_\gamma) {\rightarrow \atop \rightarrow} \prod\_{(\alpha,\beta)} F(U\_\alpha \times\_U U\_\beta)$$


(This is also what I used to prove $1) \implies 2)$). 


$\bullet$ Let $R$ be an arbitrary covering sieve, which is a subfunctor of $Hom(-,U)$. Then it contains a covering family {$U\_\alpha \to U$}, and the diagram of $2)$ is an equalizer by assumption. If $S$ is the covering sieve generated by the covering family (so $S \subset R$), the diagram above is also an equalizer. By an isomorphism of equalizers, the map $F(U) \to {{\lim \atop \leftarrow} \atop {V\to U \in S}} F(V)$ is an isomorphism. 


Because of the commutative diagram


$F(U) \to {{\lim \atop \leftarrow} \atop {V\to U \in R}} F(V)$


$\ \ \ \ \ \ \cong \searrow \ \ \ \ \ \downarrow$


$\ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\lim \atop \leftarrow} \atop {V\to U \in S}} F(V)$


we get that the map $F(U) \to {{\lim \atop \leftarrow} \atop {V\to U \in R}} F(V)$ is a split monic. From here, I'm not sure how to show that it's also epi.


So the crux of my issue is that I can prove the isomorphism in $1)$ only for covering sieves generated by covering families, but not for arbitrary covering sieves.



Equivalence of two definitions of sheaves on a site
<http://arxiv.org/abs/math/0412512> Prop 2.4.2 (sorry no time to type it myself right now).


