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Suppose $\mathcal{C}$ is a locally small category with coproducts, and there is a functor $G: \mathcal{C} \to \operatorname{Set}$ which is representable, with representation $(A, x)$. I am trying to show that $G$ has a left adjoint, and I have been given a clue that this is the functor $F : \operatorname{Set} \to \mathcal{C}$ where on objects, $F(I) = \displaystyle\coprod_{i\in I} A$, the coproduct of $|I|$ copies of $A$. However, I am not sure where a function $f:I \to J$ is mapped to by this functor. Can anyone give me an idea? Thanks for any help.

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Define $F(f)$ by mapping the $i$th copy of $A$ in $\amalg_i A$ to the $f(i)$th copy of $A$ in $\amalg_j A$. – Arturo Magidin May 10 '12 at 17:05
So I think this means the unique map $F(f) : \coprod_i A \to \coprod_j A$ such that $F(f) \circ \iota_i = \iota_{f(i)}$, where the $\iota$s are the maps into the coproduct? – Paul Slevin May 10 '12 at 17:15
Correct. Although if you want to be finicky about notation you may as well use different symbols for the different coproducts. – Zhen Lin May 10 '12 at 17:16
of course, thanks – Paul Slevin May 10 '12 at 17:23

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