Let be $\mathcal F$ a presheaf on a topological space $X$, the usual definition of the sheaf associated to $\mathcal F$ is the following
$\mathcal F^+(U)=\{\widetilde s: U\rightarrow Et(\mathcal F)\;|\; s\in \mathcal F(U),\;\textrm{and}\;\widetilde s(x)=s_x \}$
where $s_x$ is the germ at $x$ of $s$.
In the book "Algebraic geometry: an introduction - D.Perrin ", I have found another definition of the sheaf associated to a presheaf in the particular case of sheaves of functions.
Let be $\mathcal F$ a presheaf (on $X$) of functions with codomain the set $A$ then
$\mathcal F^+(U)=\{f:U\rightarrow A\;|\;\forall x\in U, \exists V\subseteq U\; \textrm{where $V$ is an open set containing $x$ and}\; \exists g\in\mathcal F(V)\;\textrm{such that}\; f|_V=g\}$
My question is the foillowing: why, in the case of sheaves of function, these definitions are the same? The nature of the elements of $\mathcal F^+(U)$ is different, infact in the first case we have function from $U$ to $Et(\mathcal F)$ instead in the second case the codomain is $A$.