Why do you want this notion of equivalence or adjunction, rather than the stricter notion of isomorphism of categories?
As far as I understand/can tell, the context of discovery in category theory is mostly category theorists noticing that a particular kind of abstract structure occurs in many different contexts and thus deserves a name. The context of justification in category theory is mostly category theorists using a particular definition in various downstream things and showing how things fit nicely, globally, everything being reflected in everything else/the primordial ooze, that sort of stuff.
To give an example, if you have a category C with all products and coproducts, you can conceive them as functors from the product category C×C to C itself. We can define a “diagonal functor”, Δ:C→C×C that just “copies” each object and morphism, ΔA=⟨A,A⟩. It turns out the coproduct is its left adjoint and the product is its right adjoint: ⊔⊣Δ⊣×.
Now, if you fix any particular object X and think of the product as an endofunctor on C, (−×X):C→C and of exponentiation as another endofunctor (−)X:C→C, then these two again form an adjunction: (−×X)⊣(−)X. Using the definition of an adjunction in terms of hom-set isomorphisms, this is just the currying thing: HomC(A×X,B)≅HomC(A,BX). In fact, this adjunction can be used as the basis to define the exponential object. For example, here’s an excerpt from Sheaves in Geometry and Logic.
As far as I understand/can tell, the context of discovery in category theory is mostly category theorists noticing that a particular kind of abstract structure occurs in many different contexts and thus deserves a name. The context of justification in category theory is mostly category theorists using a particular definition in various downstream things and showing how things fit nicely, globally, everything being reflected in everything else/the primordial ooze, that sort of stuff.
To give an example, if you have a category C with all products and coproducts, you can conceive them as functors from the product category C×C to C itself. We can define a “diagonal functor”, Δ:C→C×C that just “copies” each object and morphism, ΔA=⟨A,A⟩. It turns out the coproduct is its left adjoint and the product is its right adjoint: ⊔⊣Δ⊣×.
Now, if you fix any particular object X and think of the product as an endofunctor on C, (−×X):C→C and of exponentiation as another endofunctor (−)X:C→C, then these two again form an adjunction: (−×X)⊣(−)X. Using the definition of an adjunction in terms of hom-set isomorphisms, this is just the currying thing: HomC(A×X,B)≅HomC(A,BX). In fact, this adjunction can be used as the basis to define the exponential object. For example, here’s an excerpt from Sheaves in Geometry and Logic.