TITLE: Prove category of constructible sheaves is abelian
QUESTION [6 upvotes]: Let $X$ be a nice enough topological space, perhaps a complex algebraic variety with its analytic topology.  I'm hoping someone could help me prove that the category $\text{Constr}(X)$ of constructible sheaves on $X$ is abelian.  
I think one could use that subsheaves and quotients of local systems are local systems.  However, conceptually what is confusing me is that given two constructible sheaves $\mathcal{F}, \mathcal{G}$, they might be constructible with respect to completely different Whitney stratifications of X, right?  
So given a morphism $\mathcal{F} \to \mathcal{G}$ of constructible sheaves, can you always "refine" the two Whitney stratifications to get one such that the morphism restricts to a morphism of local systems?  
Or should I be thinking along another lines?

REPLY [12 votes]: Verdier has proved (for multiple classes of spaces, including in particular complex varieties) that for any finite set of analytic subsets, there exists a Whitney stratification for which all these analytic subsets are unions of strata.
In particular, given two constructible sheaves, and a stratification on which they are lisse, there exists a Whitney stratification that refines both of them. 
This suffices to check that kernels and cokernels of maps of constructible sheaves are constructible, as well as direct sums of constructible sheaves.