Suppose that $\mathfrak{U}$ is a covariant functor on the open sets of $X$ to $\mathbb{Z}$-modules with satisfies the following two conditions :
(a) If $U$ and $V$ are open then the sequence $$\mathfrak{U}(U\cap V)\stackrel{\alpha}{\rightarrow}\mathfrak{U}(U)\oplus\mathfrak{U}(V)\stackrel{\beta}{\rightarrow}\mathfrak{U}(U\cup V)\rightarrow 0$$ is exact, where $\alpha=i_{U,U\cap V}-i_{V,U\cap V}$ and $\beta=i_{U\cup V,U}+i_{U\cup V,V}$.
(b) If $\{U_{\alpha}\}$ is an upward-directed family of open sets with $U=\bigcup U_{\alpha}$, then $\mathfrak{U}(U)=\varinjlim \mathfrak{U}(U_{\alpha})$
How to show that $\mathfrak{U}$ is a cosheaf ?
(Exercise 3 in BREDON G. E., $\textit{Sheaf Theory}$, McGraw-Hill Book Company, New York-Toronto, Ont.-London, 1967, p. 248)
The full proof is given by Bredon in this paper page 3.
The general idea is that your assuption (a) will give you the cosheaf property for finite coverings. (You can proceed by induction, the case $n=2$ being your assumption.) Then, thanks to an inductive limit and assumption (b) you can get the cosheaf property for any covering.