There are several objects called "virtual fundamental classes." For example, certain Deligne-Mumford stacks, quasi-smooth derived schemes, etc. will admit a "perfect obstruction theory" as defined by Behrend-Fantechi and then the mojo of intrinsic normal cones produces a "virtual fundamental class" $[X] \in A_d(X)$ of virtual dimension $d$ in the Chow ring of the stack (or whatever) $X$.

Already here, there could be different perfect obstruction theories e.g. different morphisms $\varphi : E^\cdot \to \mathbb{L}_X$, $\varphi' : E'^\cdot \to \mathbb{L}_X$. How can we compare the virtual classes $[X]_{\varphi}, [X]_{\varphi'}$ coming from the two different perfect obstruction theories? Are they always the same?

Further, there are many approaches to defining virtual classes for moduli problems in symplectic geometry. There, one does not use perfect obstruction theories. Rather, one looks for "Polyfold structures" or "Kuranishi structures" which can be used to produce a homology class which is then called "the virtual fundamental class." I suppose one would like to compare the classes gotten from Polyfolds with the ones gotten from Kuranishi structure. And there are many appraoches to Kuranishi structure, so I suppose one would like to compare the virtual classes gotten from those approaches.

I apologise if there is a big well-known theorem stating asserting they are all equal. But even in that event, I think comparing the virtual classes in algebraic geometry with those in symplectic geometry is interesting.

And so, I ask: What makes these classes "virtual fundamental." Is there an axiomatic list of properties which either determines the thing or allows us to say general "model independent" things?