Let $X$ be a topological space, and $\mathcal{B} = (U_i)_i$ be a basis for its topology. Let $F := \underset{i}{\bigoplus} F_i$ be a direct sum of presheaves on $X$. We denote by $F_{\mathcal{B}}$ its restriction to $\mathcal{B}$. We can sheafify this data on $\mathcal{B}$ to get a $\mathcal{B}$-sheaf $\mathcal{F}_\mathcal{B}$. Moreover, there is an equivalence of categories between $\mathcal{B}$-sheaves and sheaves on $X$, and therefore we can construct a sheaf $\mathcal{F}$ which agree with $\mathcal{F}_{\mathcal{B}}$ on $\mathcal{B}$.
My question is the following: is $\mathcal{F}$ the sheafification of $F$, that is the direct sum (in the category of sheaves): $$\mathcal{F} = \underset{i}{\bigoplus} \mathcal{F}_i,$$ with $\mathcal{F}_i$ being the sheafification of $F_i$ ?
In other words, does the functor (which is an equivalence of categories) $$Sh(X) \to Sh(\mathcal{B})$$ from the category of sheaves on $X$ to the category of $\mathcal{B}$-sheaves preserves direct sums ?
Thanks a lot !