The concept of "pullback" has several definitions depending on the context in which it is applied, e.g., smooth functions on manifolds, differential forms, multilinear forms and so forth. See, for example, the Wikipedia Page for an enumeration of these definitions. Is there a "universal definition" of pullback from which these various specialized definitions can be derived or should these definitions be viewed as intrinsic and independent? I am aware of the categorical representation as discussed here, but I don't believe (perhaps I'm mistaken) that these various specialized definitions can be derived from the categorical one.
Also, is it a coincidence that the adjoint of an operator and the pullback operation both share the exponentiated $*$ as an indicator or is it reflective of a deeper relationship?