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

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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 17 / 3 / ii / Solution by

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

f : A \to B

gives a natural isomorphism

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

by postcomposition, with inverse

h_{f^{- 1}}

.

Conversely suppose

α : h_{A} \to h_{B}

is a natural isomorphism, with inverse

β

. Since the Yoneda embedding is a full and faithful functor, there are unique

f : A \to B

and

g : B \to A

with

h_{f} = α

and

h_{g} = β

. Their composites satisfy

h_{g f} = β α = 1_{h_{A}} = h_{1_{A}}, h_{f g} = α β = 1_{h_{B}} = h_{1_{B}} .

Faithfulness gives

g f = 1_{A}

and

f g = 1_{B}

. Thus

A ≅ B ⟺ C (-, A) ≅ C (-, B) naturally .

The naturality requirement ensures that all incoming morphisms are respected; unrelated componentwise bijections alone would not justify the conclusion.

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