Suppose the small category $\mathcal C$ is <Cartesian closed>. The <Yoneda embedding> preserves finite products. For $A,B,X\in\mathcal C$, the <Yoneda lemma> and the <exponential object> adjunction give
$$
(yB)^{yA}(X)
=\operatorname{Nat}(yX\times yA,yB)
\cong\mathcal C(X\times A,B)
\cong\mathcal C(X,B^A)
=y(B^A)(X).
$$
The bijections are natural in $X$, so
$$
\boxed{(yB)^{yA}\cong y(B^A).}
$$
Thus the essential image of $\mathcal C$ in its <presheaf topos> is full and closed under finite products and exponentials.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 119 / 6 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-119/6/b/solution