Every finite-dimensional vector space V is isomorphic to both its dual V* and its double dual V**. Which isomorphism is natural, and why?
ABoth are natural — all isomorphisms between spaces of the same dimension are natural
BNeither is natural — categorical naturality requires a group structure, not just vector space structure
CV ≅ V** is natural via the evaluation map; V ≅ V* is not natural because it requires a basis choice
DV ≅ V* is natural because duals are a canonical construction; V ≅ V** requires choosing an evaluation point
The isomorphism V → V** given by ev_V(v)(φ) = φ(v) requires no choices — it works the same way for every vector space and every linear map, and the naturality square commutes for any f: V → W. By contrast, the isomorphism V → V* sends each basis vector to its dual basis functional, which requires first choosing a basis. Different basis choices give different isomorphisms. This is precisely what 'natural' means in category theory: canonical, requiring no arbitrary choices. The V ≅ V* isomorphism exists but varies with convention.
Question 2 Multiple Choice
To verify that a natural transformation α: F ⇒ G is a natural isomorphism, what must you do?
AFind a single morphism α^{-1}: G ⇒ F at the level of entire functors, not just components
BFind an inverse isomorphism α_c^{-1}: G(c) → F(c) for each object c; naturality of the inverses is then automatic
CVerify that F and G are isomorphic as objects in the functor category, which requires a separate computation
DFind an inverse for each α_c and independently verify that the collection {α_c^{-1}} satisfies the naturality squares
The key convenience of natural isomorphisms is that you work componentwise. For each object c, find an inverse isomorphism α_c^{-1}: G(c) → F(c). Once you have these and know α is a natural transformation with each α_c an isomorphism, the collection {α_c^{-1}} automatically forms a natural transformation G ⇒ F — you do not need to separately verify naturality of the inverses. This follows from the fact that naturality squares for α can be 'inverted' using the invertibility of each component.
Question 3 True / False
If α: F ⇒ G is a natural isomorphism, then the collection of inverses {α_c^{-1}} automatically assembles into a natural transformation G ⇒ F.
TTrue
FFalse
Answer: True
This is a key feature of natural isomorphisms. If α is natural and each α_c is an isomorphism, then the inverses α_c^{-1} satisfy the naturality condition for the transformation G ⇒ F automatically. Naturality of α says α_d ∘ F(f) = G(f) ∘ α_c for any morphism f: c → d. Since each component is invertible, we can apply α_c^{-1} on the right and α_d^{-1} on the left to obtain the naturality square for the inverse transformation. You get naturality of α^{-1} for free.
Question 4 True / False
Assigning an isomorphism between F(c) and G(c) for nearly every object c in C is sufficient to make α a natural isomorphism between functors F and G.
TTrue
FFalse
Answer: False
Having an isomorphism at each component is necessary but not sufficient. The collection {α_c} must also be natural — meaning for every morphism f: c → d in C, the square α_d ∘ F(f) = G(f) ∘ α_c must commute. Without naturality, you merely have a collection of unrelated isomorphisms, not a coherent relationship between functors. The naturality condition is what ensures the identification 'respects the structure of C and D' and makes F and G categorically interchangeable.
Question 5 Short Answer
What does it mean for two functors to be naturally isomorphic, and how does this differ from having isomorphisms between their values at each object?
Think about your answer, then reveal below.
Model answer: Two functors F, G: C → D are naturally isomorphic if there is a natural transformation α: F ⇒ G where each component α_c: F(c) → G(c) is an isomorphism, and these components commute with all morphisms in C. Having isomorphisms at each object (without naturality) only says the values are individually related; naturality ensures the isomorphisms are coherent — they transform consistently with the structure of C. Naturally isomorphic functors are categorically interchangeable: any categorical statement about F applies equally to G.
The distinction captures what 'canonical' means in mathematics. A choice of isomorphisms at each object might depend on arbitrary conventions (like a choice of basis), giving a different isomorphism in each context. Naturality rules this out: the isomorphisms must fit together consistently with every morphism, making them independent of choices. This is why 'naturally isomorphic' is the correct notion of sameness for functors, just as 'isomorphic' is the correct notion for objects.