$0 \rightarrow A \xrightarrow{i} B \xrightarrow{\pi} C \rightarrow 0$

Where the first arrow is $i$ for inclusion and the third is $\pi$ for projection.
We now want to consider what happens when we have $Hom(X, -)$ for some target space $X$.
The induced arrows are induced from composition; I will write $(f; g)(x) \equiv g(f(x))$
to mean "first do $f$, then do $g$". Hence, my arrows linking $Hom(X, A)$ to $Hom(X, B)$ will be
$-;i$: To first go from $X$ to $A$, and then apply $i$ to go from $A$ to $B$.
This gives us the sequence (which we are to check if it is exact, and which of the left
and right arrow exist):
$0 \xrightarrow{?} Hom(X, A) \xrightarrow{-;i} Hom(X, B) \xrightarrow{;-\pi} Hom(X, C) \xrightarrow{?} 0$

$0 \xrightarrow 2Z \rightarrow{\pi} Z \rightarrow{\pi} Z/2Z \rightarrow 0$

We now have three interesting choices for our $X$ in relation to the above sequence:
(a) $Z$, (b) $2Z$, (c) $Z/2Z$. Since $Z$ and $2Z$ are isomorphic as modules, let's study
the case with $Z$ and $Z/2Z$