A simple example would be the following: Suppose you have a secure MAC scheme $M=(K,T,V)$ with the keygen, tagging and verification algorithms. Then define the new scheme $M'=(K,T',V')$ with $T'(k,m)=T(k,m)\|s$ for some constant string $s$ and $V'$ checking with a reject for the presence of $s$ before returning whatever $V$ would have returned on the shortened input.
The security reduction for $M'$ would then usually go as follows: You're given an adversary $\mathcal A$ against the sEUF-CMA security of $M'$ and need to construct $\mathcal R$ running the sEUF-CMA experiment to break $M$ using $\mathcal A$ as a sub-routine.
For any MAC'ing query from $\mathcal A$ you would then forward and on the way backwards, you'd add $s$ at the end of the tag.
For any Verifying query from $\mathcal A$ you then check if $s$ is present at the end of the tag and if not, immediately respond with the reject and otherwise forward the stripped tag-message pair also forwarding the result.
In the end, if $\mathcal A$ wins, it needs to have sent you a valid verify query which passed forwarded verification which also means the reduction adversary $\mathcal R$ has won.
