# Equivalence between two definitions of sheaves

I am reading "Notes on Grothendieck topologies, fibered categories and descent theory", version of october 2, 2008, from Angelo Vistoli, he introduces the definition of sheaf on page $31$, definition $2.37$, item $ii$:

Definition $2.37$. Let $\mathcal{C}$ be a site, $F:\mathcal{C}^{op}\rightarrow (Set)$ a functor.

$(i)$ $F$ is separated if, given a covering ${\mathcal{U}_i\rightarrow\mathcal{U}}$ and two sections $a$ and $b$ in $F\mathcal{U}$ whose pullbacks to each $F\mathcal{U}_i$ coincide, it follows that $a=b$.

$(ii)$ $F$ is a sheaf if the following condition is satisfied:suppose that we are given a covering ${\mathcal{U}_i\rightarrow\mathcal{U}}$ in $\mathcal{C}$, and a set of elements $a_i\in F\mathcal{U}_i$. Denote by $pr_1:\mathcal{U}_i\times_{\mathcal{U}}\mathcal{U}_j\rightarrow \mathcal{U}_i$ and $pr_2:\mathcal{U}_i\times_{\mathcal{U}}\mathcal{U}_j\rightarrow \mathcal{U}_j$ the first and the second projection respectively, and assume that $pr_{1}^{*}a_i=pr_{2}^{*}a_j\in F(\mathcal{U}_i\times_{\mathcal{U}}\mathcal{U}_j)$ for all $i$ and $j$. Then there is a unique section $a\in F\mathcal{U}$ whose pullback to $F\mathcal{U}_i$ is $a_i$ for all $i$.

If $F$ and $G$ are sheaves on a site $\mathcal{C}$, a morphism of sheaves $F\rightarrow G$ is simply a natural transformation.

A sheaf on a site is separated.

I want to prove the equivalence of that definition with the one given on category theory, using equalizers:

Let $\mathcal{C}$ be a Grothendieck site. A pre-sheaf of Abelian groups is a functor $F:\mathcal{C}^0\rightarrow Ab$.

A sheaf is a pre-sheaf such that for every object $\mathcal{U}$ of $\mathcal{C}$ and every covering ${\mathcal{U}_i\rightarrow \mathcal{U}}$, the following diagram

$F(\mathcal{U})\xrightarrow{Equalizer} \prod_iF(\mathcal{U}_i)\xrightarrow[g]{f}\prod_{i,j} F(\mathcal{U}_i\times_{\mathcal{U}}\mathcal{U}_j)$

is an equalizer."

Where the definition of equalizer is the following:

Definition of equalizer: Let $X,Y\in\mathcal{C}$ and $f,g\in Mor(E,X)$. An equalizer is a pair $(E,Eq)$ such that $E\in\mathcal{C}$ and $f\circ Eq=g\circ Eq$ and if there is another pair $(A,h)$ such that $A\in \mathcal{C}$ and $h\in Mor(A,X)$ such that $f\circ h= g\circ h$, then $\exists!\Phi:A\longrightarrow E$ with $Eq\circ \Phi=h$.

This is pretty quick once you know the constructions of the diagram you're taking the equalizer of and of equalizers in the category of sets. For the first, the map $f$ is precisely $f((a_i))=(pr_1^*a_i)$ and similarly for $g$. The equalizer is this exactly all the families $(a_i)$ such that $pr_1^*a_i=pr_2^*a_j$ for every $i$ and $j$.