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Trying to understand the equivalence between the different notions of a sheaf

I have the following problem:
I would like to prove that the notions of sheaf and presheaf are equivalent; moreover, that sheaves of sets and presheaves of sets are equivalent. But I am unsure of how to prove that the following proof holds:
Let $U \subset X$ be an open subset of $X$, then the set of sections of $X|_U$ is just $I^o(U)$ and then, by right adjointness of $\mathscr{C}$, $I^o(U)$ can be seen as an object of $\mathscr{C}$: $I^o(U) \cong \mathscr{C}(U)$. Therefore by right adjointness of $V \mapsto \mathscr{C}(V)$ for open sets $V \subset X$, we can equivalently define $X \mapsto \mathscr{C}$ by $X \mapsto \{f: X \to \mathscr{C}: f \text{ is a function and } f|_U \in \mathscr{C}(U)\}$.
So I believe we have a functor, and that the easiest way to prove that the two notions of sheaf are equivalent is to show that two functors have the same image

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