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 $Eq\circ \Phi=h$.

Suggestions and answers will be tremendously appreciated. Thanks in advance.


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 $j$.


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