= Solution
Here a <pointed complete partial order> is a <partially ordered set> $D$ with a least element $\bot_D$ in which every nonempty <directed set> has a <least upper bound>. Directed means that every pair of its elements has an upper bound within that set. A <Scott continuous map> is monotone and preserves these directed suprema. It is not required to preserve the least element; restricting to strict maps would change the categorical claim. If completeness is instead formulated only for increasing countable chains, the same arguments below work with those chains.
Order the <function space of complete partial orders> $[D\to E]$ pointwise: $f\leq g$ means $f(x)\leq g(x)$ for every $x$. Its least element is the constant function with value $\bot_E$. For a directed family $\mathcal F$ of continuous maps, define $h(x)=\sup_{f\in\mathcal F}f(x)$. This is well-defined and is its pointwise <least upper bound>. It is monotone. For any directed $A\subseteq D$,
$$
h(\sup A)=\sup_f f(\sup A)=\sup_f\sup_{a\in A}f(a)=\sup_{a\in A}\sup_f f(a)=\sup h[A].
$$
The interchange is valid because the combined family $\{f(a):f\in\mathcal F,a\in A\}$ is directed, and either iterated supremum is its <least upper bound>. Thus $h$ is a <Scott continuous map>, proving that \b[$[D\to E]$ is a <pointed complete partial order>].
Finite products carry the <product order>, componentwise suprema and least element $(\bot_D,\bot_E)$. The one-element order is a <terminal object>. For the exponential, use the function space with evaluation $\operatorname{ev}(f,x)=f(x)$. To check joint continuity, take a directed family of pairs $(f_i,x_i)$. Then
$$
\operatorname{ev}(\sup_i f_i,\sup_i x_i)=\sup_i\sup_j f_i(x_j)=\sup_k f_k(x_k).
$$
Indeed, directedness supplies a $k$ above the pairs indexed by both $i$ and $j$, giving $f_i(x_j)\leq f_k(x_k)$. Thus the diagonal values are cofinal in the double family, and evaluation preserves its supremum.
For a continuous $h:A\times D\to E$, its <currying> is $\Lambda h:A\to[D\to E]$, $(\Lambda h)(a)(d)=h(a,d)$. Each section is continuous because $d\mapsto(a,d)$ is; pointwise directed suprema in the $a$ coordinate show that $\Lambda h$ is continuous. Conversely a continuous $g:A\to[D\to E]$ gives the continuous map $\operatorname{ev}\circ(g\times\operatorname{id}_D)$. The two constructions are inverse, because they agree on every pair $(a,d)$. This gives the natural bijection
$$
\boxed{\operatorname{Hom}(A\times D,E)\cong\operatorname{Hom}(A,[D\to E]),}
$$
with the required evaluation map. Together with the <terminal object> and products, it proves \b[cartesian closure of pointed complete partial orders with non-strict continuous maps].
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