Many definitions of $2$-categories are given as categories equipped with some extra structure encoded by some functors and some natural (or extranatural) transformations between these functors (or with $Id$ functors, etc.), satisfying some commutativity conditions between them.
Sometimes, one of those "equipments" are of natural isomorphism, and sometimes one wishes to relax to a more general $2$-category where this is no longer a natural isomorphism, but only one direction is given, while retain some of the original theory (having a left adjoint to the inclusion of the $2$-categoires can give quite a lot of the original structure).
Is there a cannonial process to convert the axioms of the theory with the natural isomorphism to a more general theory, where we require only one direction of the isomorphism - but with a description of the same form: A list of commutativity of some diagrams given by the "extra structure"?
The generalized $2$-category given by this description should probably be definied via some universal property, I presume.
My motivation comes from a very specific example: I am working with Street's Skew closed categories, but it seems that for one construction to work, that one of the commutative diagrams in the definition of a skew-closed category be reversed (and this is due to the non-invertibility of $i$ which is a natural isomorphism in more restrictive $2$-category). So I am wondering, if such a process exists, and Skew closed categories are "the right generalization", then this means my construction is "not the right one".