  Every maximally monotone operator can be associated with a family of convex
functions, called the Fitzpatrick family or family of representative functions.
Surprisingly, in 2017, Burachik and Mart\'inez-Legaz showed that the well-known
Bregman distance is a particular case of a general family of distances, each
one induced by a specific maximally monotone operator and a specific choice of
one of its representative functions. For the family of generalized Bregman
distances, sufficient conditions for convexity, coercivity, and supercoercivity
have recently been furnished. Motivated by these advances, we introduce in the
present paper the generalized left and right envelopes and proximity operators,
and we provide asymptotic results for parameters. Certain results extend
readily from the more specific Bregman context, while others only extend for
certain generalized cases. To illustrate, we construct examples from the
Bregman generalizing case, together with the natural "extreme" cases that
highlight the importance of which generalized Bregman distance is chosen.
