In MacLane's "Categories for the Working Mathematician", in the end of section VII.1 there is an argument due to Isbell proving that one cannot strictify a monoidal category by replacing it with its skeleton.

Generally, I would say that all one can achieve by replacing a category with its skeleton is replacing natural isomorphisms with much less natural automorphisms.

In MacLane's "Categories for the Working Mathematician", in the end of section VII.1 there is an argument due to Isbell proving that one cannot strictify a monoidal category by replacing it with its skeleton.

Generally, I would say that all one can achieve by replacing a category with its skeleton is replacing natural isomorphisms with much less natural automorphisms.

In MacLane's "Categories for the Working Mathematician", in the end of section VII.1 there is an argument due to Isbell proving that one cannot strictify a monoidal category by replacing it with its skeleton.

Generally, I would say that all one can achieve by replacing a category with its skeleton is replacing natural isomorphisms with much less natural automorphisms.

In MacLane's "Categories for the Working Mathematician", in the end of section VII.1 there is an argument due to Isbell proving that one cannot strictify a monoidal category by replacing it with its skeleton.

Generally, I would say that all one can achieve by replacing a category with its skeleton is replacing natural isomorphisms with much less natural automorphisms.