Lemma 7.26.4. Let $\mathcal{C}$ be a site. Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. Given any glueing data $(\mathcal{F}_ i, \varphi _{ij})$ for sheaves of sets with respect to the covering $\{ U_ i \to U\} _{i \in I}$ there exists a sheaf of sets $\mathcal{F}$ on $\mathcal{C}/U$ together with isomorphisms

\[ \varphi _ i : \mathcal{F}|_{\mathcal{C}/U_ i} \to \mathcal{F}_ i \]

such that the diagrams

\[ \xymatrix{ \mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j} \ar[d]_{\text{id}} \ar[r]_{\varphi _ i} & \mathcal{F}_ i|_{\mathcal{C}/U_ i \times _ U U_ j} \ar[d]^{\varphi _{ij}} \\ \mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j} \ar[r]^{\varphi _ j} & \mathcal{F}_ j|_{\mathcal{C}/U_ i \times _ U U_ j} } \]

are commutative.

**Proof.**
Let us describe how to construct the sheaf $\mathcal{F}$ on $\mathcal{C}/U$. Let $a : V \to U$ be an object of $\mathcal{C}/U$. Then

\[ \mathcal{F}(V/U) = \{ (s_ i)_{i \in I} \in \prod _{i \in I} \mathcal{F}_ i(U_ i \times _ U V/U_ i) \mid \varphi _{ij}(s_ i|_{U_ i \times _ U U_ j \times _ U V}) = s_ j|_{U_ i \times _ U U_ j \times _ U V} \} \]

We omit the construction of the restriction mappings. We omit the verification that this is a sheaf. We omit the construction of the isomorphisms $\varphi _ i$, and we omit proving the commutativity of the diagrams of the lemma.
$\square$

## Comments (3)

Comment #1183 by JuanPablo on

Comment #1184 by Johan on

Comment #1185 by JuanPablo on