Suppose we have an exact sequence of $R$-modules
\begin{array}{ccccccccc} 0 & \longrightarrow & L & \overset{f}{\longrightarrow} & M & \overset{g}{\longrightarrow} & E & \longrightarrow & 0\\ \end{array} Let $\phi : L \to L'$ be a homomorphism of $R$-modules.
We know that there is a bijection between the ( classes of isomorphic ) extensions $e(E,L)$ and $\text{Ext}_1(E,L)$.
Moreover Ext$_1(-,-)$ is a functor contravariant in the first component and covariant in the second. My question is: what is the concrete morphism of extensions induced by $$\text{Ext}_1(E,\phi) : \text{Ext}_1(E,L) \to \text{Ext}_1(E,L')$$ ? i.e. what are the maps which define a commutative diagram
$$\begin{array} 00 & {\longrightarrow} & L & \stackrel{f}{\longrightarrow} & M \stackrel{g}{\longrightarrow} & E & {\longrightarrow} & 0\\ & & \downarrow{\alpha} & & \downarrow{\beta} & \downarrow{id} \\ 0 & {\longrightarrow} & L' & \stackrel{f'}{\longrightarrow} & M' \stackrel{g'}{\longrightarrow} & E & {\longrightarrow} & 0\\ \end{array} $$
?