Module isomorphic to a flat module Let $M$ be a flat $A$-module, and $N$ a $A$-module isomorphic to $M$, what can we say about the flatness of $N$? 
 A: Let $f:M\to N$ be an isomorphism of $A$-modules, and let $P$ be an arbitrary $A$-module. Then $P\times M\to P\otimes_A N$, $(p,m)\mapsto p\otimes f(m)$ is $A$-bilinear, hence we get an induced well-defined homomorphism
$$\operatorname{id}\otimes f:P\otimes_A M\to P\otimes_A N, p\otimes m\mapsto p\otimes f(m).$$
In the same way, we have an inverse homomorphism $\operatorname{id}\otimes f^{-1}$, such that $P\otimes_A M\cong P\otimes_A N$.
Now $M$ being flat means that if $g:P\to P'$ is an injective morphism of $A$-modules, $g\otimes\operatorname{id}:P\otimes_A M\to P'\otimes_A M$ is, too. But then the map
$$P\otimes_AN\xrightarrow{\sim}P\otimes_AM\xrightarrow{g\otimes\operatorname{id}}P'\otimes_AM\xrightarrow{\sim}P'\otimes_A N$$
is an injective $A$-homomorphism, which proves the flatness of $N$.
As for your question regarding Georges' answer, I'm not exactly sure what you mean. In general, one defines these properties such that they stay invariant under isomorphism in the respective category. But looking for a proof of this, as Georges calls it, meta-rule, wouldn't really make sense to me.
