It is a basic fact of mathematics that given $r$ maps $f_i: X_{i-1}\to X_i$ $\ (1\leq i\leq r)$ one can apply these maps in turn to arbitrary points $x\in X_0$ and gets as output a point $y\in X_r$ which is defined by $y=f_r\Bigl(f_{r-1}\bigl(\ldots f_2(f_1(x))\ldots\bigr)\Bigr)\ .$ In this way a map $\phi:\quad X_0\to X_r, \quad x\mapsto y$ is generated, and one writes $\phi:=f_r\circ f_{r-1}\circ\ldots\circ f_1$.
This $\phi$ is a well-defined mathematical object by itself. As it leads directly from $x\in X_0$ to $y\in X_r$ the individual $f_i$ will no longer be visible in its expression (and can in fact be discarded). The actual expression of $\phi$ depends on circumstances. When, e.g., all $f_i$ are linear (or affine) maps then $\phi$ is such a map as well and can be expressed by a matrix (plus a constant vector). When the $f_i$ are defined in more complicated ways the resulting expression for $\phi$ will be more complicated, too. But in any case the expression of $\phi$, taking the encoding of an arbitrary $x\in X_0$ as input (e.g., $x=(x_1,x_2,x_3)$) and giving the encoding of the image point $y:=\phi(x)$ as output, can be computed once and for all.