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?
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 $g=\operatorname{Hom}_{\mathscr D}(X,h)\circ\eta_S$.