This might be a highly trivial question since I’ve just encountered category theory for a short time. Let $A$ be a commutative ring with identity and consider the polynomial ring $A[x]$ as the free $A$-algebra generated by a single element $x$. If we consider the construction of $A[x]$, then for any ring homomorphism $h: A\to B$ we have a naturally induced algebra homomorphism $h^*:A[x]\to B[x]$ defined by $$h^*(\sum\limits_{i=0}^n a_ix^i)=\sum\limits_{i=0}^n h(a_i)x^i.$$ Moreover if $h: A\to B$ preserves units,(that is, $h(u)$ is a unit in $B$ if $u$ is a unit in $A$) then $h^*$ also preserves units.
The above statements are not just useless information. For example consider the following proposition in elementary commutative ring theory;
Let $A$ be a commutative ring with identity. If the polynomial $f=\sum\limits_{i=0}^n a_ix^i$ is a unit in $A[x]$ then $a_0$ is a unit in $A$ and $a_1,...,a_n$ are nilpotent elements in $A$.
We utilize the claims from the first paragraph to reduce the proposition to integral domains. For every prime ideal $\mathfrak{p}\subset A$ consider the unit-preserving canonical homomorphism $h: A\to A/\mathfrak{p}$. Assume the proposition is true for integral domains $A/\mathfrak{p}$. Since the induced map $h^*: A[x]\to (A/\mathfrak{p})[x]$ preserves units, all coefficients for $h^*(f)$ except for the constant term are $0$ in $A/\mathfrak{p}$ and thus coefficients $a_1,...,a_n$ for $f$ belong to every prime ideal of $A$, which means that $a_1,...,a_n$ are all nilpotent elements. $a_0$ is easily seen to be a unit in $A$.
After all the above informations, I have two questions:
Can be obtain the definition for the induced map $h^*$ without looking at the construction? Note that this works in general for any free object: There is a functor from a concrete category $(C, F)$ to the category of free objects generated by one(or any positive integer $n$?) element. Intuitively it is just defined by mapping an element in an object $O\in C$ to itself, but in the content of free objects, and maps a homomorphism in $(C, F)$ to its induced homomorphism between free objects.
Note that for many propositions like above in commutative algebra we are able to obtain both “global” and “local” proofs.(For example the “local” version of proof to the above proposition is just listing inductively cancel out the elements.) Is there a general explanation for why this phenomenon happens? I think mostly it is because we can nicely obtain a “global” characterization for the set of special elements. For example the set of units in a commutative ring with identity is the complement of the union of all its maximal ideals, the nilradical is the intersection of all its prime ideals, etc.