Let $\mathcal{C}$ be a category with coproducts.

Show that if $G:\mathcal{C} \to \mathbf{Set}$ is representable then $G$ has a left adjoint.

I can't seem to wrap my head around this nor why coproducts are required.

By definition $G$ is naturally isomorphic to some hom functor $\mathcal{C}(X,-)$ but I don't know where to go from here.

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    $\begingroup$ Do you know any examples of categories $C$ and functors $G$ satisfying these hypotheses? Have you tried playing around with them? $\endgroup$ – Qiaochu Yuan May 3 '13 at 21:51
  • $\begingroup$ (I just wanted to mention that Qiaochu's comment has a great educational value as compared to a complete answer which just writes down the left adjoint) $\endgroup$ – Martin Brandenburg May 4 '13 at 8:22

Hint: If $F$ is left adjoint to $G$ (just assume that it exists for the moment), then $F(*)=X$, and $F$ preserves colimits, in particular coproducts. Now compute $F(S)$ for an arbitrary set $S$. After that, show that, in fact, $F$ defined by ... exists and is left adjoint to $G$.


Hint: Let $G=\hom(X,-)$. Consider $F:Set\to C$ by $S\mapsto \cup_S X$ ($S$ copies of $X$).


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