Questions: Composition of Functors and Functor Equations

Question 1

Given functors F: A → B, G: B → C, and H: C → D, which of the following correctly describes the relationship between (HG)F and H(GF)?

• A. They are naturally isomorphic but not necessarily equal as functors
• B. They are strictly equal as functors — associativity of functor composition holds on the nose
• C. They are equal only when all categories involved are small
• D. They are equal only when F, G, and H are all full and faithful

Answer: (B)

Functor composition is strictly associative — (HG)F = H(GF) as literal equalities, not merely up to natural isomorphism. This follows because functors are just functions on objects and morphisms satisfying extra laws, and composition of functions is strictly associative. No coherence isomorphism is needed. This strict equality (as opposed to 'up to isomorphism') is precisely what gives Cat the structure of a strict 2-category rather than a bicategory.

Question 2

To verify that the composite GF: A → C preserves composition — that (GF)(g ∘ f) = (GF)(g) ∘ (GF)(f) — the key step is:

• A. Showing that G and F have compatible object assignments on shared objects
• B. Applying F's preservation of composition to get F(g ∘ f) = F(g) ∘ F(f), then applying G's preservation of composition to that result
• C. Showing that GF is injective on morphisms
• D. Applying the naturality condition for the identity natural transformation

Answer: (B)

The proof chains the two functor laws: first apply F's composition law to rewrite F(g ∘ f) = F(g) ∘ F(f), then apply G's composition law to rewrite G(F(g) ∘ F(f)) = G(F(g)) ∘ G(F(f)) = (GF)(g) ∘ (GF)(f). The proof is two applications of the functor axiom in sequence — no other properties of F or G are needed. This pattern of 'apply the inner functor law, then the outer' recurs throughout category theory.

Question 3

Functor composition is associative mainly up to natural isomorphism — that is, (HG)F and H(GF) are naturally isomorphic but may not be literally equal.

• T. True
• F. False

Answer: False

Functor composition is strictly associative: (HG)F = H(GF) as an equality of functors, not merely an isomorphism. This distinguishes Cat (the category of small categories with strict associativity) from a bicategory (where associativity holds only up to coherent isomorphism). The strict equality holds because functor composition is defined pointwise via function composition, which is itself strictly associative.

Question 4

The identity functor id_A: A → A satisfies F ∘ id_A = F = id_B ∘ F for any functor F: A → B, making it a strict unit for composition.

• T. True
• F. False

Answer: True

The identity functor sends every object and every morphism to itself. Pre-composing any functor F: A → B with id_A gives a functor that maps each object a to F(id_A(a)) = F(a), and each morphism f to F(id_A(f)) = F(f) — exactly F itself. Similarly for post-composing with id_B. These equalities hold strictly, not just up to isomorphism, for the same reason associativity holds strictly.

Question 5

Explain why the strict (not merely 'up to isomorphism') associativity of functor composition is significant for the structure of Cat as a 2-category.

Model answer: In a strict 2-category, the composition of 1-morphisms (here: functors) must be strictly associative and have strict units — actual equalities, not just coherent isomorphisms. Cat satisfies this because functor composition is defined via pointwise function composition, which is strictly associative. If associativity held only up to natural isomorphism, Cat would be a bicategory (weak 2-category), requiring additional coherence data (associator and unitor isomorphisms satisfying pentagon and triangle identities). The strict equality simplifies the theory: proofs about functor composition can treat parenthesization as irrelevant rather than tracking coherence isomorphisms. This strictness is the algebraic foundation for treating Cat as a category of categories, with functors as its morphisms.

The distinction between strict and weak 2-categories becomes important when working with structures like monoidal categories or 2-functors, where weakening associativity introduces nontrivial coherence requirements. Understanding that Cat is strict clarifies when such coherence data is truly needed versus when it can be suppressed.