If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of torsion free coherent sheaves on a surface. Here $M$ is a line bundle, $E$ a vector bundle of rank 2 and $N$ the quotient. The exact sequence corresponds to the Harder Narasimhan filtration of $E$ with respect to an ample $L$. So we have that $\mu_L(M)>\mu_L(E)>\mu_L(N)$.

In the paper I am looking at, they consider the Moduli stack of extensions of the above type, call this $\mathcal{E}$. There is a natural projection $p:\mathcal{E}\longrightarrow\mathcal{M}(v(M))\times\mathcal{M}(v(N))$. Here $\mathcal{M}(v(M))$ denotes the moduli space of coherent sheaves on $S$ with mukai vector $v$. They say that the fiber over a point $(M,N)$ is the quotient $[\textrm{Ext}^1(N,M)/\textrm{Hom}(N,M)]$?

How does the quotient make sense? What is the action of $Hom(N,M)$ on Ext$^1(N,M)$. They further say that action is trivial. But I am not able to understand what the action is at all.

So how do we get this quotient? Thanks in advance!