 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-18/2/d/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

α : F \Rightarrow G

[C, Set]

β : C (A, -) \Rightarrow G

β

corresponds to

x = β_{A} (1_{A}) \in G (A)

y \in F (A)

with

α_{A} (y) = x

γ_{B} (f) = F (f) (y)

C (A, -) \Rightarrow F

α

yields

α_{B} γ_{B} (f) = α_{B} F (f) (y) = G (f) (α_{A} y) = G (f) (x) = β_{B} (f) .

Therefore covariant representables are projective in the set-valued functor category. Local smallness ensures that

C (A, -)

is set-valued.

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