C

is Cartesian closed. The Yoneda embedding preserves finite products. For

A, B, X \in C

, the Yoneda lemma and the exponential object adjunction give

(y B)^{y A} (X) = Nat (y X \times y A, y B) ≅ C (X \times A, B) ≅ C (X, B^{A}) = y (B^{A}) (X) .

The bijections are natural in

X

, so

(y B)^{y A} ≅ y (B^{A}) .

Thus the essential image of

C

in its presheaf topos is full and closed under finite products and exponentials.

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