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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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    @PaulSlevin Maybe you want to answer your own question with the hints in the comments, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-15

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