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Is the product of two objects in a concrete category always a sub-object of the product of these objects in the category of sets, i.e. is their product in the concrete category a subset of their cartesian product?

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    The word "is" here rubs me the wrong way. I'd rather you rephrase the question as follows: for $F : C \to \text{Set}$ any functor, there's a natural morphism in $\text{Set}$ from $F(X \times Y)$ to $F(X) \times F(Y)$ given by the universal property of $F(X) \times F(Y)$. The question is whether this morphism is a monomorphism if $F$ is faithful.2012-02-27

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