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?