I've found this claim:
Let $\mathcal{C}$ be a category; the family $\lbrace C_i \rbrace_{i \in I }$ has a coproduct in $\mathcal{C}$ if and only if the functor $$F : \mathcal{C} \to Set$$ $$A \mapsto \bigcup_{i \in I } \hom(C_i , A )$$ is representable.
Is it right ? Why there isn't the functor $$A \mapsto \prod_{i \in I} \hom(C_i, A )\ ?$$