**Edit of Feb. 14, 2019.** After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is

Do binary coproducts always exist in the category of noetherian commutative rings?

**End of edit.**

I asked the question on Mathematics Stackexchange but got no answer.

Let $\mathsf{Noeth}$ be the category of noetherian rings, viewed as a full subcategory of the category $\mathsf{CRing}$ of commutative rings with one.

Let $A$ be in $\mathsf{CRing}$.

Question 1.Is there a functor from a small category to $\mathsf{Noeth}$ whose limit in $\mathsf{CRing}$ is $A$?

Let $f:A\to B$ be a morphism in $\mathsf{CRing}$ such that the map $$ \circ f:\text{Hom}_{\mathsf{CRing}}(B,C)\to\text{Hom}_{\mathsf{CRing}}(A,C) $$ sending $g$ to $g\circ f$ is bijective for all $C$ in $\mathsf{Noeth}$.

Question 2.Does this imply that $f$ is an isomorphism?

Yes to Question 1 would imply yes to Question 2.

Question 3.Does the inclusion functor $\iota:\mathsf{Noeth}\to\mathsf{CRing}$ commute with colimits? That is, if $A\in\mathsf{Noeth}$ is the colimit of a functor $\alpha$ from a small category to $\mathsf{Noeth}$, is $A$ naturally isomorphic to the colimit of $\iota\circ\alpha$?

Yes to Question 2 would imply yes to Question 3, and yes to Question 3 would imply that many colimits, and in particular many binary coproducts, do *not* exist in $\mathsf{Noeth}$: see this answer of Martin Brandenburg.

Here are two particular cases of the above questions:

Question 4.Is $\mathbb Z[x_1,x_2,\dots]$ a limit of noetherian rings?

(The $x_i$ are indeterminates.)

Question 5.Do binary coproducts exist in $\mathsf{Noeth}$?

One may try to attack the first question as follows:

Let $A$ be in $\mathsf{CRing}$ and $I$ the set of those ideals $\mathfrak a$ of $A$ such that $A/\mathfrak a$ is noetherian. Then $I$ is an ordered set, and thus can be viewed as a category. We can form the limit of the $A/\mathfrak a$ with $\mathfrak a\in I$, and we have a natural morphism from $A$ to this limit. I'd be interested in knowing if this morphism is bijective. [Edit. By a comment of Laurent Moret-Bailly the morphism in question is not always bijective.]

limits, not colimits. In question 1 I wrote "Is there a functor from a small category to $\mathsf{Noeth}$ whose limit in $\mathsf{CRing}$ is $A$?". Do you think it's not a clear enough definition of what is meant by "limit"? $\endgroup$ – Pierre-Yves Gaillard Feb 13 at 13:10