Currying in a locally small category with coproducts While studying for category theory course I stumbled upon the following question taken from a previous exam:

Let $\mathcal{D}$ be a locally small category with all coproducts. Show that for every object $X \in \mathcal{D}$, the functor $(−) \times X : \mathcal{Set} \rightarrow D$ is left adjoint to $\operatorname{Hom}_{\mathcal{D}} (X, −)$.

I have some troubles with this question and am quite convinced that it is not well defined. 

The first thing that bugged me was that the question implies that $(−) \times X$ is a functor from $\mathcal{Set}$, which in my understanding should be a functor from $\mathcal{D}$.
But even assuming that this was a typo, I am still confused of why the existence of coproducts is required, when taking a product is... well, a product.

Concretely, my question is:

Can anyone 'guess' how the question was actually intended and sketch the main ideas of how to prove this?

 A: Let $S$ be a set and $X$ be an object of $\mathscr C$.
Then $S\times X$ can be regarded as
$$\coprod_{s\in S}X$$
that's the coproduct indexed by elements of $S$ of copies of the object $X$.
Note that the symbol $\times$ in $S\times X$ doesn't denote a product in $\mathscr D$.
This notation come from the fact that coproduct is often denoted as a sum, hence in our case we get (informally):
$$\sum_{s\in S}X=\underbrace{X+\cdots+X}_S=S\times X$$
Now let $\kappa_s:X\to S\times X$ for $s\in S$ denote the inclusion morphism into that coproduct.
Then
\begin{align}
&\eta_S:S\to\operatorname{Hom}_{\mathscr D}(X,S\times X)&
&s\mapsto\kappa_s
\end{align}
is the unit of the adjunction.
For if $Y$ is another object in $\mathscr D$ and $g:S\to\operatorname{Hom}_{\mathscr D}(X,Y)$ is a function, then we have a cocone $g(s):X\to Y$ for $s\in S$.
By universal property of coproducts, there exists one and only one morphism $h:S\times X\to Y$ such that $g(s)=h\circ\kappa_s$ for every $s\in S$ and this means $g=\operatorname{Hom}_{\mathscr D}(X,h)\circ\eta_S$.
