Let $\mathbb{K}$ denote a class of structures, in the usual sense of model theory. Have there been any attempts to define a sensible notion of either
the usual/canonical siganture with which to axiomatize $\mathbb{K}$,
a "category of all signatures suitable to axiomatize $\mathbb{K}$".
If so, what references do you recommend on this?
Remarks.
A standard example of two different signatures being used to axiomatize the same class is the class of all 1-categories: there is of course the usual two-sorted axiomatization, and the less usual but well-known one-sorted one.
The "syntactic category", in the usual sense of categorical logic, is vaguely in the spirit of 2., but seems quite irrelevant actually.
I have not seen anything substantial in the direction of either 1. or 2. There seems no way, and it seems not be considered a reasonable question.