Label Invariance with respect to $f_1, f_2$ (Definition) Suppose ($A^2, p_1, p_2:A^2\rightarrow A$) is a binary product of objects $A\ and \ A$ in an arbitrary category $C$. Suppose $f_1, f_2:T\rightarrow A$ are maps in $C$. By definition of product,
$$\exists! f:T\rightarrow A_2 \ such \ that \ p_1\circ f=f_1 \wedge p_2\circ f=f_2.$$ We call definition of product label invariant with respect to $f_1, f_2$, if $\exists f' \ such \ that \ p_1 \circ f'=f_2 \wedge p_2\circ f'=f_1$. Because, otherwise two people can disagree on whether an object is a product if one labelled $f_1, f_2$ opposite to other person. Hence, the name label invariance.
My question:
Assuming the definition of a product as " In category $C$, an object $P$ together with maps $p_1:P\rightarrow A_1, p_2:P\rightarrow A_2$ is called product of objects $A_1, A_2$ if $\forall \ objects \ T \in C, \forall f_1:T\rightarrow A_1, \forall f_2:T\rightarrow A_2, \exists! f:T\rightarrow P$ such that $p_1\circ f=f_1 \wedge p_2\circ f=f_2$, if one can prove the label invariance, how?
Secondly, I think, this points at my ignorance of some general concept at play. So, I request you to be as general as possible.
There's nothing hard here at all. Suppose $(P,p_1,p_2)$ is a product of objects $A_1=A$ and $A_2=A$. Now suppose that $g_1:T\to A$ and $g_2:T\to A$ are two maps (these are the maps called $f_1$ and $f_2$ when proving label invariance, but I'm giving them different names to avoid confusion). Our goal is to prove that there exists $f'$ such that $p_1\circ f'=g_2$ and $p_2\circ f'=g_1$.
How do we prove this? Well, we just apply the definition of a product, with $f_1=g_2$ and $f_2=g_1$. These maps $f_1$ and $f_2$ are indeed maps $T\to A_1=A$ and $T\to A_2=A$, so the definition applies and says there exists a unique $f:T\to P$ such that $p_1\circ f=f_1=g_2$ and $p_2\circ f=f_2=g_1$. So, if we take $f'=f$, we're done.
The key point here is that the definition of a product says "for all $f_1$ and $f_2$". So in that definition, $f_1$ and $f_2$ are just placeholder variables, which could refer to any two maps $T\to A_1$ and $T\to A_2$ at all. In particular, you could let $f_1$ refer to the map "$f_2$" in the statement of label invariance and let $f_2$ refer to the map "$f_1$" in the statement of label invariance. (But, to avoid horribly confusing yourself, I recommend renaming one of these sets of variables, like I did with $g_1$ and $g_2$ above.)