I think this is a silly question, but I'm having troubles with the abstraction in category theory. Let $\mathcal{C}$ be a category and $A,B$ two objects in $\mathcal{C}$. Prove that if $A \times B$ exists, then $B \times A$ exists.
Now, $A \times B$ exists means that $A \times B$ is an object of $\mathcal{C}$ and there are two arrows $\pi_1 :A \times B \rightarrow A$, $\pi_2: A \times B \rightarrow B$ such that for any object $S$ in $\mathcal{C}$ and for any arrows $f_1: S \rightarrow A$, $f_2: S \rightarrow B$ there exists a unique arrow $u: S \rightarrow A \times B$ such that $f_1 = \pi_1 \circ u$, $f_2 = \pi_2 \circ u$.
Let $S'$ be an object of $\mathcal{C}$ and $f_1': S' \rightarrow A$, $f_2': S' \rightarrow B$. In order to prove that $B \times A$ exists I have to prove that this is an object of $\mathcal{C}$ and there are two arrows $p_1: B \times A \rightarrow B$, $p_2: B \times A \rightarrow A$ such that there's a unique $u': S' \rightarrow B \times A$ with $f_1'=p_2 \circ u'$, $f_2'=p_1 \circ u'$. But for that $(S',f_1',f_2')$ I know that $f_1'= \pi_1 \circ u$. And now I'm stuck.
Any help please? Thanks
Best Answer
It may help if you avoid the names $A \times B$ and $B \times A$ of the objects that 'exist'.
You know there is some object $P$ with two arrows $\pi_1 \colon P \to A$ and $\pi_2 \colon P \to B$ satisfying the universal property of the product of $A$ and $B$. You have to find some object $Q$ with two arrows $p_1 \colon Q \to B$ and $p_2 \colon Q \to A$ satisfying the universal property of the product of $B$ and $A$. So, you just take $Q = P$, $p_1 = \pi_2$ and $p_2 = \pi_1$ and everything works out just fine.