There’s the equivalence of categories. Two categories are equivalent when they are isomorphic up to an isomorphism. Specifically, if you have two functors F:C→D and G:D→C, such that there are natural isomorphisms (invertible natural transformations) α:F∘G∼→1D and β:1C∼→G∘F. On objects, this means that if you start at an object X in C, then you can go to FX in D and then to GFX in C that is isomorphic to X: GFX≅X. Similarly if you start at some object in D.
Equivalence is an isomorphism when the isomorphisms GFX≅X (equivalently,[1] the natural transformations between the compositions and identity functors) are equalities.
An even weaker equality-like notion is adjunction and this where things start to get asymmetrical. There’s a few equivalent[2] ways of defining them[3], but the contextually simplest way (if not completely rigorous), since I just described equivalences, is that it’s an equivalence, except the natural transformations α:F∘G→1D and β:1C→G∘F are not (in general) isomorphic. So, you go XF→FXG→GFX and you could instead have gotten there via some morphism X→GFX in the category C but you may not be able to go back GFX→X. On the other hand, starting at some Y in D, you can take the trip YG→GYF→FGY and go back to where you started via some morphism within D, FGY→Y. Again, there may not be a morphism Y→FGY.
Then we say that F is left adjoint to G (equivalently, G is right adjoint to F), denoted F⊣G. The natural transformations ηX:X→GFX and ϵY:FGY→Y are called the unit and the counit of the adjunction, respectively.
Seven Sketches in Compositionality introduces adjunctions in a slow and palatable way that is good for building an intuition, starting with Galois connections, which are just adjunctions for preorders, which are just Bool-categories.
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.
There’s the equivalence of categories. Two categories are equivalent when they are isomorphic up to an isomorphism. Specifically, if you have two functors F:C→D and G:D→C, such that there are natural isomorphisms (invertible natural transformations) α:F∘G∼→1D and β:1C∼→G∘F. On objects, this means that if you start at an object X in C, then you can go to FX in D and then to GFX in C that is isomorphic to X: GFX≅X. Similarly if you start at some object in D.
Equivalence is an isomorphism when the isomorphisms GFX≅X (equivalently,[1] the natural transformations between the compositions and identity functors) are equalities.
An even weaker equality-like notion is adjunction and this where things start to get asymmetrical. There’s a few equivalent[2] ways of defining them[3], but the contextually simplest way (if not completely rigorous), since I just described equivalences, is that it’s an equivalence, except the natural transformations α:F∘G→1D and β:1C→G∘F are not (in general) isomorphic. So, you go XF→FXG→GFX and you could instead have gotten there via some morphism X→GFX in the category C but you may not be able to go back GFX→X. On the other hand, starting at some Y in D, you can take the trip YG→GYF→FGY and go back to where you started via some morphism within D, FGY→Y. Again, there may not be a morphism Y→FGY.
Then we say that F is left adjoint to G (equivalently, G is right adjoint to F), denoted F⊣G. The natural transformations ηX:X→GFX and ϵY:FGY→Y are called the unit and the counit of the adjunction, respectively.
Seven Sketches in Compositionality introduces adjunctions in a slow and palatable way that is good for building an intuition, starting with Galois connections, which are just adjunctions for preorders, which are just Bool-categories.
no pun intended
double no pun intended
many such cases in category theory
Thaaaaat is a confusing sentence. But thankfully, the rest of your comment clears things up.
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.