  We consider a type of dependent percolation introduced by Aizenman and
Grimmett, who showed that certain "enhancements" of independent (Bernoulli)
percolation, called essential, make the percolation critical probability
strictly smaller. In this paper we first prove that, for two-dimensional
enhancements with a natural monotonicity property, being essential is also a
necessary condition to shift the critical point. We then show that (some)
critical exponents and the scaling limit of crossing probabilities of a
two-dimensional percolation process are unchanged if the process is subjected
to a monotonic enhancement that is not essential. This proves a form of
universality for all dependent percolation models obtained via a monotonic
enhancement (of Bernoulli percolation) that does not shift the critical point.
For the case of site percolation on the triangular lattice, we also prove a
stronger form of universality by showing that the full scaling limit is not
affected by any monotonic enhancement that does not shift the critical point.
