Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For $n \geq m$, let $\varphi_{m,n}$ denote the canonical ring homomorphism $R / I^n \to R / I^m$. Let $J = \cap_n I^n$. Then $R / J$ is going to end up being the $I$-adic completion of $R$, coming equipped with the obvious natural maps $\lambda_n \colon R / J \to R / I^n$.
My question regards a certain step in establishing that this object is final in the relevant category. Suppose $S$ is a commutative ring coming equipped with maps $\varphi_n \colon S \to R / I^n$ such that for each $n$, $\varphi_{n, n+1} \circ \varphi_{n+1} = \varphi_n$. We want to show that there exists a unique ring homomorphism $\varphi \colon S \to R / J$ such that for each $n$, $\varphi_n$ factorss through $R / J$ appropriately via $\varphi$.
Now, the definition of $\varphi$ is forced on us. Note that for a fixed $s \in S$ and arbitrary $n$, the assumptions on $S$ imply that the coset $\varphi_{n+1}(s)$ of $I^{n+1}$ is contained in the coset $\varphi_{n}(s)$ of $I^n$. So the $\varphi_n(s)$ form a nested sequence of cosets of the corresponding powers of $I$, and it is straightforward to see that the only candidate for $\varphi(s)$ is
$$\varphi(s) = \bigcap_n \varphi_n(s).$$
Assuming that the set on the right is nonempty (moreover, assuming it contains a coset of $J$), it is immediate that it is a coset of $J$: for each $r,r' \in \bigcap_n \varphi_n(s)$, $r - r' \in I^n$ for each $n$, and so $r - r' \in J$.
But why is it guaranteed that $\bigcap_n \varphi_n(s)$ contains a coset of $J$? Intuitively it seems this must be true: the "size" of all the cosets of $J$ are the same. Nevertheless, it seems to me that some work has to be done; for instance, we are aware that the question of the intersection of infinitely many nested sets is a delicate question, say, in topology, where the nonempty intersection of sequences of nested closed sets is equivalent to the ambient space being compact. In this problem, I am concerned because we have made no assumptions on the ring $R$ (for example, if $R$ is Noetherian then this is trivially true).
As a note, this formulation of the $I$-adic completion is taken from Aluffi's text on algebra. And of course, this leads to the construction of the $p$-adic numbers.
Brandon Carter's response in the comments suffices.