# 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?

• The question is stated accurately. Notice that $\text{Hom}_\mathcal D(X,Y)$ is a set. The described functor is called a copower (or also tensor but "tensor" is massively overloaded) and is usually written with $\otimes$. – Derek Elkins left SE Mar 17 '19 at 1:27

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