The lower shriek functor is defined by
$$f_{!}F(U)=\{s\in\Gamma(f^{-1}(U),F)\;:\; f|_{\mathrm{supp}(s)}:\mathrm{supp}(s)\rightarrow U\text{ is proper}\}$$
On the other hand, if $j:V\subset X$ is the inclusion of an open set, the extension by zero functor is defined by $$ j_{!}F(U)=\begin{cases} F(jU)=F(U) & U\subset V\\ 0 & \text{otherwise} \end{cases}$$
How can I prove these definitions coincide for inclusions of open sets? I don't know anything about base-change theorems, so I'd like to avoid them.