The hom-set bijection Hom_D(F(A), B) ≅ Hom_C(A, G(B)) must satisfy what additional condition beyond being a bijection of sets?
AIt must be defined only for finite categories
BIt must be natural in both A and B
CIt must be the identity function when A = B
DIt must send identity morphisms to identity morphisms
A mere bijection that varies erratically with A and B would not capture a structural relationship. Naturality means the bijection is preserved under morphisms in both variables — this coherence is what makes the adjunction a structural fact about the categories rather than an accidental correspondence.
Question 2 True / False
If F ⊣ G is an adjunction (F left adjoint to G), then F and G together form an equivalence of categories.
TTrue
FFalse
Answer: False
An equivalence requires both the unit η: 1_C → GF and counit ε: FG → 1_D to be natural isomorphisms. An adjunction only requires them to satisfy the triangle identities without being isomorphisms. Many important adjunctions (e.g., free group ⊣ forgetful) are not equivalences.
Question 3 Short Answer
In what sense does the free–forgetful adjunction between Set and Grp illustrate the hom-set bijection?
Think about your answer, then reveal below.
Model answer: A function from a set S to the underlying set U(G) of a group G corresponds naturally to a group homomorphism from the free group F(S) to G; this bijection Hom_Grp(F(S), G) ≅ Hom_Set(S, U(G)) is natural in both S and G.
The free group construction provides exactly the structure needed to uniquely extend any set-function to a group homomorphism. This is the hom-set bijection in action: morphisms out of the free object on the left correspond perfectly to structure-preserving maps on the right.