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I'm attending the 6th World Congress and School on Universal Logic. The first tutorial that I'm attending is being given by Henri Prade, A logical view of analogical reasoning based on analogical proportions. He's describing analogies using logical relationships, for example, comparing the set relationships A-B and C-D.
Analogies are of the form: "A is to B as C is to D".
I'm thinking that category theory is natural for modeling analogies. If in one category we have a morphism m:A->B, then the question is whether there is a functor F such that F(m):F(A)->F(B) where we call F(A)=C and F(B)=D. Sometimes the analogy/functor exists and sometimes not. Sometimes the analogy/functor can only be defined in one way and sometimes not. But it seems that analogies capture the commutative diagrams that we draw with functors and morphisms.
Perhaps you have thought about this or might be interested to discuss this.