I am slowly realizing that my understanding of category theory is shaky. Thus, I have the two following (very basic) question.
In the category $\mathbf{Sets}$, is $A \times B$ an object of the category, with $A$ and $B$ actually objects of the category (i.e., sets)?
It seems to me the alternative would be to explicitly introduce a product category $\mathbf{Sets} \times \mathbf{Sets}$, whose objects are of the form $A \times B$, with $A$ and $B$ sets.
(Small aside, if $A \times B$ is indeed an object of $\mathbf{Sets}$, its relation to the final object $\mathbf{1}$ in $\mathbf{Sets}$ (a singleton set) provided by $! : A \times B \to \mathbf{1}$ is as expected?)
I have the feeling the answer to my question is positive and it is related to the fact that $\mathbf{Sets}$ is a Cartesian closed category (which should also cover the finality issue, I think), but I am not really sure.
Thanks a lot for your feedback and your time!