$\widetilde{F}=\bigoplus\limits_{x\in X}F_{\{x\}}$ is not isomorphic to $F$

Question

I want to find a sheaf $F$ and a topological space $X$ such that $\widetilde{F}=\bigoplus\limits_{x\in X}F_{\{x\}}$ is not isomophic to $F$.
Where $F_{\{x\}}$ is the skyscraper sheaf, this is \begin{align} F_{\{x\}}(U)=\begin{cases}F_x \quad\text{if}\quad\ U\ni x,\\0 \quad\text{else}. \end{cases} \end{align}

Note that $\widetilde{F}_y=F_y$ for all $y\in X$.

Any hit would be appreciated.

Answer 1

As user10354138 suggested if we consider $F$ the constant sheaf $\mathbb{Z}/2\mathbb{Z}$ on the indiscrete two-point space $X=\{x,y\}$ we have the following:

$\widetilde{F}(X)=F_{\{x\}}(X)\oplus F_{\{y\}}(X)=F_x\oplus F_y=\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}.$

While, $F(X)=\{s:X\rightarrow \mathbb{Z}/2\mathbb{Z}:\text{s is locally constant}\}=\mathbb{Z}/2\mathbb{Z}.$

Is this correct?