I am doing an exercise in Qing Liu's Algebraic Geometry and Arithmetic Curves on an alternative definition of the sheafification of a presheaf but can't see why the gluing property holds.
More precisely, let $\mathcal{F}$ be a presheaf (of rings say) on a topological space $X$. Fix a cover $\mathcal{U}=\{U_i\}$ of $X$ then we get a complex $$0 \rightarrow \mathcal{F}(X) \overset{d_0}\rightarrow \prod\limits_i\mathcal{F}(U_i) \overset{d_1}\rightarrow \prod\limits_{i,j} \mathcal{F}(U_{ij}),$$ where $U_{ij} =U_i \cap U_j$, $d_0(s)=(s|_{U_i})_i$ and $d_1((s_i)_i)=(s_i|_{U_{ij}}-s_j|_{U_{ij}})_{i,j}$.
Define $\mathcal{F}_{\mathcal{U}}(X)=\ker d_1$ and extend to all open sets $V \subset X$ with the cover $\{U_i \cap V\}$ to define the presheaf $\mathcal{F}_{\mathcal{U}}$.
By refining our cover of $X$, we get a directed set and define $\mathcal{F}'(V)= \lim\limits_{\rightarrow} \mathcal{F}_{\mathcal{U}}(V)$ to be the direct limit across the covers.
I want to show that this is actually a sheaf but can't seem to verify the gluing axiom. I know that this should actually be the sheafification and all the stalks are therefore equal but I don't see how to use this.
Any help would be much appreciated.