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?

Whitneystratification? I've been confused whether you need Whitney in the definition of constructible, because I've seen it both ways. $\endgroup$