 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-17/3/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

A

h_{A} = C (-, A)

and

u : A \to B

to postcomposition

h_{u} : h_{A} \to h_{B}

. Apply the Yoneda lemma with

X = h_{B}

Nat (h_{A}, h_{B}) ≅ h_{B} (A) = C (A, B) .

The inverse sends

u

precisely to

h_{u}

, because its component at

C

f : C \to A

u f

. Hence the induced maps on all hom-sets are bijections. The Yoneda embedding is full and faithful.

 New to topics? Read the docs here!