Let $X$ be a topological space and $\{U_i\}$ and open cover for $X$. Suppose we have sheaves $\mathcal{F}_i$ on $U_i$ and for each $i,j$ an isomorphism $\varphi_{ij} : \mathcal{F}_i|_{U_i \cap U_j} \to \mathcal{F}_j|_{U_i \cap U_j}$ such that $\varphi_{ii} = id$ and $\varphi_{ik} = \varphi_{jk} \circ \varphi_{ij}$ on $U_i \cap U_j \cap U_k$. I want to define some kind of sheaf $\mathcal{F}$ on $X$; to do this we consider

$$\mathcal{F}(W) := \Big\{(s_i)_{i \in I}, s_i \in \mathcal{F}_i(W \cap U_i) : \varphi_{ij}(s_i|_{W \cap U_i \cap U_j}) = s_j|_{W \cap U_i \cap U_j} \Big\}.$$

My question is:Can we realise $\mathcal{F}(W)$ as some kind of limit of a diagram, or yet as an equalizer of two maps? I ask this because I want to show that this $\mathcal{F}$ is a sheaf. At the moment it seems to me that $\mathcal{F}(W)$ is very close to being some kind of ``inverse limit", but I don't know exactly what it is.