= Solution
The <fan of the affine line> consists of the zero cone and $\mathbb R_{\geq0}e'$. Under $\varphi(e_1)=0$ and $\varphi(e_2)=e'$, the four primitive source rays map as
$$
e_1+e_2\mapsto e',\qquad
e_1\mapsto0,\qquad
e_2\mapsto e',\qquad
-e_1\mapsto0.
$$
Every cone of the source fan consequently maps into $\mathbb R_{\geq0}e'$. The defining criterion for a <toric morphism> is satisfied, so $\varphi$ induces
$$
f:X_\Sigma\longrightarrow\mathbb A^1.
$$
Under the blowup description from part (i), this is the blowup map followed by the projection $\mathbb P^1\times\mathbb A^1\to\mathbb A^1$.
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