Relation between representable functors and adjunction

Question

In class we proved that given two adjoint functors $F\dashv G:\mathbb C\to \bf{Set}$, where the category $\mathbb C$ is arbitrary, $G$ is (naturally) isomorphic to $H^{F*}$, where $*$ is the singleton (as an object of $\bf {Set}$). Instead, to prove that, given a $G:\mathbb C\to \bf{Set}$ represented by the object $C$, there is a left adjoint $F\dashv G$, we noticed that a left adjoint preserves the colimits. So given a set $I$, it can be written as $\coprod_{i\in I} (*)_i$, viewing it as the disjoint union of singleton; since $F$ must preserve coproducts, we try to define it as $I\mapsto \coprod_{i\in I} (C)_i$. Now, I should first $F$ also on arrows, and then prove that this is a left adjoint. However, I don't see how to extend $F$, and moreover I don't understand how to prove immediately that this is a left adjoint (the teacher said that this last check is very easy to do). So this should be an easy exercise, but I am stuck with it; thanks in advance for any clarify.

Answer 1

If I understand your question correctly, you are considering a category $\mathcal C$, an object $c \in\mathcal C$ and want to show that (if $\mathcal C$ admits all the necessary coproducts), $G = \operatorname{Hom}_{\mathcal C}(c,-)$ has a left afjoint $F$ given by $Fx = \coprod_x c $.

How to define $F$ on arrows

Let $f:x\to y$ be an arrow in $\mathbf{Set}$. For each $j \in y$, there is a canonical monomorphism $\iota_j: c \hookrightarrow \coprod_y c$. Using the $\iota_{f(i)}$ for each $i \in x$ and the universal property of the coproduct, you can define a morphism $Ff: \coprod_x c \to \coprod_y c$. It is the only morphism such that : $$\forall i \in x,F f\circ\iota_i = \iota_{f(i)}$$ Using this last property, it is easy to show that this makes $F$ into a functor.

Proof that $F \dashv G$

For $x\in\mathbf{Set}$ and $y \in \mathcal C$, you have the following isomorphisms (natural in $x$ and $y$) : \begin{align*} \operatorname{Hom}_{\mathcal C} (\coprod_x c,y) &\simeq \prod_x \operatorname{Hom}_{\mathcal C}(c,y) \\ &\simeq \prod_x G(y) \\ &\simeq \operatorname{Hom}_{\mathbf{Set}}(x,G(y)) \end{align*}

The first line is the universal property of the coproduct, the second the definition of $G$ and the last is the definition of the cartesian product in $\mathbf{Set}$.