Here is the question I am trying to understand its solution:
Show that the maps $G \xrightarrow{n} G$ and $H \xrightarrow{n} H$ multiplying each element by the integer $n$ induce multiplication by $n$ in $\operatorname{Ext}(H,G).$
Here is the solution I found online:
My questions are:
1- In the line after the first diagram, what is the value of $\operatorname{im}f_{1}^{*}$? why the author said that the multiplication by $n$ in $G$ induces multiplication by $n$ in $\operatorname{Hom}(F_1, G)$ only, why he did not speak about the denominator, $\operatorname{im}f_1^{*}$?
2- Why in the first diagram the induced $n^{*}$ is pointing downwards while in the third diagram the induced $n^*$ is pointing upwards?
3- What about the induced map between $\operatorname{Hom}(F_0, G)$ and $\operatorname{Hom}(F, G)$ in the first diagram?
Can someone clarify this to me please?