If $L$ is a toric line bundle on $X_{\Sigma_{2}}$ and $\varphi$ is a toric morphism, then $\varphi^{\ast}L$ has an induced toric structure. Namely, $\varphi^{\ast}(L,z)=(\varphi^{\ast}L,\varphi^{\ast}z)$. By contrast, if $\varphi\colon X_{\Sigma_{1}}\to X_{\Sigma_{2}}$ is a general equivariant morphism that meets the principal open subset, there is no natural toric structure on $\varphi^{\ast}L$, because the image of the distinguished point $x_{1,0}$ does not need to agree with $x_{2,0}$. If $(L,s)$ is a toric line bundle equipped with a toric section, then we set $\varphi^{\ast}(L,s)=((\varphi^{\ast}L,(\varphi^{\ast}s)(x_{1,0})),\varphi^{\ast}s)$. However, the underlying toric bundle of $\varphi^{\ast}(L,s)$ depends on the choice of the toric section.