= Solution
A <Cartesian closed category> has finite <categorical products> and, for every $P,Q$, an <exponential object> $Q^P$ with an evaluation <morphism> $\operatorname{ev}:Q^P\times P\to Q$ such that composition with evaluation gives <natural bijections>
$$
\mathcal E(R,Q^P)\cong\mathcal E(R\times P,Q).
$$
Finite <categorical products> in the <presheaf category> are pointwise; in particular its <terminal object> is the constant singleton <categorical presheaf>. For <categorical presheaves> $P,Q$ define
$$
\boxed{(Q^P)(c)=\operatorname{Nat}(H_c\times P,Q).}
$$
For $u:d\to c$, restriction sends $\alpha$ to $\alpha\circ(H_u\times1_P)$. Composition and identity laws follow from those of the <Yoneda embedding>, so this is a <categorical presheaf>.
Define evaluation at $c$ by $\operatorname{ev}_c(\alpha,x)=\alpha_c(1_c,x)$. For $u:d\to c$, <naturality> of $\alpha$ gives $Q(u)\alpha_c(1_c,x)=\alpha_d(u,P(u)x)$, which is the evaluation of the restricted pair. Hence evaluation is a <natural transformation>.
Given $t:R\times P\to Q$, its <currying> sends $z\in R(c)$ to the transformation whose component at $a$ is
$$
(\widehat t_c(z))_a(v,x)=t_a(R(v)z,x)\qquad(v:a\to c,\ x\in P(a)).
$$
<Naturality> in $a$ follows from that of $t$, and <naturality> in $c$ follows by composing $v$ with the relevant arrow. Evaluating $\widehat t$ at $(1_c,x)$ recovers $t_c(z,x)$. Conversely, if $s:R\to Q^P$, <naturality> of $s$ along $v:a\to c$ says that evaluating $s_a(R(v)z)$ at $(1_a,x)$ gives $(s_c(z))_a(v,x)$. Thus <currying> the evaluation of $s$ recovers $s$. These constructions are inverse and natural, proving the <universal property>. This is the <exponential in a presheaf category>, and proves \b[the <presheaf category> is <Cartesian closed>].
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