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).



> 
> **Question:**
> 
> 
> 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".



Relaxing a natural isomorphism to a natural transformation to obtain a more general $2$-category
One situation in which this can be done if the original structure can be described as a pseudo-algebra structure for some 2-monad. In this case, to make the constraints noninvertible one can consider instead lax algebras or colax algebras for the same 2-monad. However, this process tends to produce only [unbiased](http://ncatlab.org/nlab/show/bias) structures, whereas skew-things are generally biased (see for instance [here](https://ncatlab.org/nlab/show/lax+monoidal+category)), so it seems unlikely to solve your specific problem.


