Let $f : A \rightarrow B$ be a morphism of commutative rings. Let $F$ be an element of $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$. Show that we have $f \left( F \left( a _ { 1 } , \ldots , a _ { n } \right) \right) = F \left( f \left( a _ { 1 } \right) , \ldots , f \left( a _ { n } \right) \right)$ for all $a _ { 1 } , \ldots , a _ { n } \in A$.