In Bhatt's paper *On the Direct Summand Conjecture and its Derived Invariant*, he makes the following "almost" definition for pro-A-modules:

DefinitionA pro-$A$-module $\{M_n\}_{n\ge1}$ is said to be almost-pro-zero if for any $k\ge0$ and any $n\ge1$, there exists some $m = m(n,k) ≥ n$ such that $im(M_m → M_n)$ is killed by $t^{1/{p^k}}$; equivalently, for each $k ≥ 0$, the map $\{M_n[t^{1/p^k}]\}_{n≥1} \to \{M_n\}_{n≥1}$ is a pro-isomorphism in the usual sense. A map of pro-objects in $D^b(A)$ is said to be an almost-pro-isomorphism if the cohomology groups of cones form an almost-pro-zero system.

My problem seems to be that I'm ignorant on the definition of a pro-$A$-module and morphisms of these objects in general. The only seemingly relevant reference I can find is the following page on nlab

https://ncatlab.org/nlab/show/pro-object

which defines for any category $C$ a "pro-category" $\text{pro-}C$ where the objects are diagrams of the form $F:D\to C$ for $D$ a small cofiltered category, and morphisms are defined by

$$\text{pro-}C(F,G)=\varprojlim_{e\in E}\varinjlim_{d\in D} C(Fd,Ge)$$

for $F:D\to C$ and $G:E\to C$. This definition isn't very enlightening but I assume pro-$A$-modules are basically just this construction with $D$ taken to be $\Bbb N^{\rm{op}}$ (considered as a category) for all objects, meaning that objects of pro-$A$-mod are just sequences

$$\cdots\to M_{n+1}\to M_n\to M_{n-1}\to\cdots\to M_1$$

of morphisms of $A$-modules. I still feel unenlightened as to what the (iso)morphisms of these objects are, and why the remark Bhatt makes in the second statement of the quoted definition holds true.