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i need a reference of the following proof, i hope someone can show me a book where i can find it.

Theorem:

Let $U$ be a cover of a top space $X$ and $F_i$ a sheaf on $U_i$. If there exists a morphism of sheaves like $\theta_{ij}: F_i\mid_{U_{ij}} \longrightarrow F_j\mid_{U_{ij}}$, then there exists a sheaf $F$ on $X$ such that $\theta_i : F\mid_{U_{ij}} \longrightarrow F_i$ is an isomorphism.

i could not find a proof for that on my own, hopefully someone can help me

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    You need an extra hypothesis : the cocyle condition $\theta_{jk}\theta_{ij}=\theta_{ik}$. You can look at these notes for example : http://math.uchicago.edu/~chonoles/expository-notes/hartshorne-2.1.22.pdf – 2017-02-28
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    Thank you very much Roland. How is it that i have not found that theorem in the most common sheaf theory books? – 2017-03-01
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    In most books, it is left as an exercise :p (Kashiwara-Shapira, Hartshorne,...) – 2017-03-01
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    so i can basically not find the proof of that theorem in a published book? :D. thats not so good, because i need that one for my thesis – 2017-03-02

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