Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 119 / 5 / i / Solution

A Cartesian closed category has finite products in a category and, for every $F$, an exponential object $G^F$ characterized naturally byIn $[\mathcal{C},\mathbf{Set}]$, finite products and the terminal functor are computed pointwise. With $h_B=\mathcal{C}(B,-)$, defineThis is a set because $\mathcal{C}$ is a small category. For $v:B\to B^{\prime }$, precomposition $h_{B^{\prime }}\to h_B$ induces $(G^F)(v)$ by precomposition of natural transformations, making $G^F$ a covariant functor.

Given $\alpha :K\times F\to G$, define its curry $\hat{\alpha }:K\to G^F$ byFor $a:C\to C^{\prime }$, naturality of $\alpha$ gives $G(a){\alpha }_C(K(u)y,x)={\alpha }_{C^{\prime }}(K(au)y,F(a)x)$, proving that this is a natural transformation $h_B\times F\to G$. The same formula with $u$ precomposed proves naturality of $\hat{\alpha }$ in $B$.

Conversely, from $\beta :K\to G^F$ defineNaturality of $\beta$ and its values proves naturality of $\alpha$. Currying this result recovers $\beta$, since ${\beta }_C(K(u)y)=(G^F)(u)({\beta }_B(y))$; uncurrying a curry recovers $\alpha$ by setting $u=1_B$. These mutually inverse constructions are natural in $K,G$, giving the adjunction required for an exponential object. Therefore