TITLE: Relation between two distributions expressed in terms of their CDFs
QUESTION [1 upvotes]: Not great at stats, and having trouble wrapping my mind around this.  Would love an explanation, not overly detailed, in plain english of what these transformations mean.

The bias correction strategy assumes that the model discrepancies stay
  constant in time [i.e., the relationship between the distributions of
  $X_o$ and $X_m$ is the same as the relationship between the distributions of
  $X_o′$ and $X_m′$ (Figs. 1a,b)]. This allows predictions of future
  observables to be obtained by mapping from future model simulations,
  $X_o′ = B(X_m′ )$, with a transfer function given by $B(X′_m) = F_o^{-1}[F_m(X_m′)]$, where $F_m( \cdot )$ is the cumulative distribution function (CDF) of $X_m$
  and $F_o^{-1}( \cdot )$ is the inverse CDF (the “quantile function”) of $X_o$.

Image that goes along with it:

Mostly can't wrap my mind around this part:

$B(X′_m) = F_o^{-1}[F_m(X_m′)]$, where $F_m( \cdot )$ is the cumulative distribution function (CDF) of $X_m $   and $F_o^{-1}( \cdot )$ is the inverse CDF (the “quantile function”) of $X_o$.  

How do you apply a CDF from one variable to another, and then also apply the CDF to the result?  I thought these (the CDFs) were merely the characteristics of a variable, not functions that could be applied to another variable.
I should add, I understand there description, but not how their math/symbolation matches there description. 
Much thanks.

REPLY [2 votes]: CDF is a function (which is what the letter F is for), and any reasonable function $\varphi$ can be applied to a random variable $X$, producing another random variable $\varphi(X)$. As a special case, applying the CDF of $X$ to $X$ itself results in a random variable uniformly distributed between $0$ and $1$, no matter what the original distribution was (as long as it was continuous). Another way to state this: $X = \varphi^{-1}(Z)$ where $Z$ is uniformly distributed between $)$ and $1$, and $\varphi$ is the CDF of $X$. This is essentially a restatement of what CDF is. 
But here we are not really applying one CDF, but rather the composition $F_o^{-1}\circ F_m$, called the transfer function. What does this function tell us about the relation of $X_0$ and $X_m$? If $y=F_o^{-1}\circ F_m(x)$, then $F_o(y)=F_m(x)$, so the probability of $X_0\le y$ is the same as the probability of $X_m\le x$. In other words, this composition $F_o^{-1}\circ F_m(x)$ converts numbers from the range of $X_m$ to the range of $X_0$, so that the result of conversion is exactly the same percentile of $X_0$ as the original number was for $X_m$. 
What good is this for us? Well, if we have another pair of variables $X_0'$ and $X_m'$, which we believe to be in the same relation as $X_0$ and $X_m$, we can use the transfer function to predict $X_0'$ based on $X_m'$. This is the idea of plugging $X_m'$ into the transfer function.
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