I'm trying to understand the following extract from a book.
Let $coord$ be a mapping from set $D \to D'$. Let's use the notation $D_{d'}$ to indivate the representing block of $d'$, i.e., the subset of $D$ containing the elements of $D$ mapping to $d' \in D'$ through $coord$.
Given a set of maps
$h_{d'} {:} \times _{d\in D_{d'}} Q_d \to \times _{d'\in D'} Q'_{d'}$,
we require that the global map
$h {:} \times _{d \in D} Q_d \to Q'_{d'}$
to be composed of local functions $h_{d'}$.
Conceptually, I kind of get it. It is trying to suggest that the global map $h$ is "made up" of the multiple local functions $h_{d'}$. However, is the word composite proper in this context?
I guessed that composition of functions was only defined when the image of the inner-most function was contained in the domain of the outer-most function. In this case, the image of any of the $h_{d'}$ is part of the $Q_{d'}$ set, so it is not possible to apply any other $h_{d'}$ as these operate on $Q_d$ with $d \in D_{d'}$.
Am I misunderstanding something? What would be the proper operation to "stitch" the local maps $h_{d'}$ to obtain $h$?