Let $A$ be a local noetherian ring, $M$ an $A$-module finitely generated. Let $f$ be an $A$-regular and $M$-regular element (i.e. $f$ is not a zero divisors on $A$ nor on $M$). Then inside the category of $A/fA$-modules (I think we can suppose finitely generated), we have the following isomorphisms of functors:
Ext$^n_{A/fA}(M/fM,\_)\cong$ Ext$^n_A(M,\_)$ for every $n\geq0$
Ext$^n_{A/fA}(\_,M/fM)\cong$ Ext$^{n+1}_A(\_,M)$ for every $n\geq0$
I found this theorem into some notes but I don't know how to prove it. I'm even not sure to understand what it means, it seems to imply that the functor Ext$^n_A(M,\_)$ is defined on the category of $A/fA$-modules. Could you tell me how to prove it and could you help me to understand better the statement, please?