given a finite dimensional quiver algebra $A$ and a generator $M$ with $Ext^1(M,M)=0$. By Wakamatsus lemma, for any $A$-module $N$ there exists a surjective $A$-linear map $f:M_1 \rightarrow N$ with $Ext^1(M,K)=0$, where $K$ denotes the Kernel of $f$ and $M_1 \in add(M)$ (see chapter 7 of the book by Enochs and Jenda). My question is the following: What is the best/fastest way to find such a map $f$? It is known in general (when $M$ is just an arbitrary module) how to find $f$ but this is very time consuming (and Wakamatsus lemma doesnt hold in general) and I wonder whether there is a trick when $M$ is a generator with $Ext^1(M,M)=0$.

given a finite dimensional quiver algebra $A$ and a generator $M$ with $\operatorname{Ext}^1(M,M)=0$. By Wakamatsus lemma, for any $A$-module $N$ there exists a surjective $A$-linear map $f\colon M_1 \rightarrow N$ with $\operatorname{Ext}^1(M,K)=0$, where $K$ denotes the Kernel of $f$ and $M_1 \in \operatorname{add}(M)$ (see chapter 7 of the book by Enochs and Jenda). My question is the following: What is the best/fastest way to find such a map $f$? It is known in general (when $M$ is just an arbitrary module) how to find $f$ but this is very time consuming (and Wakamatsus lemma doesnt hold in general) and I wonder whether there is a trick when $M$ is a generator with $\operatorname{Ext}^1(M,M)=0$.