From roughly 1981 into 1983 I worked on "universal relations" --- a failed thesis topic which formulated isomorphism, natural maps, and cryptomorphism in terms of set theory with class-many ur-elements. Intuitively, an ur-element is a "point" with no internal structure. Two objects in this universe are isomorphic if there exists a bijection between their points which carries one to the other. Any permutation of the class of all points defines a symmetry of the universe. It is then a simple theorem that two objects are isomorphic if there exists a symmetry of the universe that maps one to the other. A natural map is a function on the universe that commutes with symmetries of the universe. Two objects x and y are cryptomorphic if there exists natural maps f and g such that f(x) = y and g(y) = x.
This formulation does not involve types --- everything is defined on individual objects relative to symmetries of the universe. It fails to properly handle the type "multiset-of(alpha)" or "bag-of(alpha)" which I now know is property modeled as the dependent pair type pair-of(beta:type, f:beta->alpha). I was aware of this weakness at the time, but did not see how to fix the problem.
At this time I developed a distinct distaste for category theory. Category theory treats objects themselves as points. This is just cognitively wrong.
While I think my early work in this area was well motivated by clear cognitive phenomenon, the work had no audience and should have been formulated in terms of type theory rather than universe symmetries. However, a proper type-theoretic formulation is nontrivial. The thesis failed. A thesis on the Ontic verification system was accepted in 1987.
The universal relation work is described in MIT AI Lab Memo 710 from 1983 Symmetric Set Theory, A General Theory of Isomorphism, Abstraction and Representation
Much later I pressured Patrick Winston into including a chapter on this work in a collection published by MIT Press in 1991.