This question may be too broad. Under what conditions is a product of noninvertible morphisms invertible?
Suppose that we model a finite number of different acts of observation (i.e., a thermometer reading, a spectrogram, a personality test, etc.) by morphisms $f_i : S\rightarrow Q_i$ (for $1 < i < I$), where $S$ represents "the real world", the source of all possible observations, and $Q_i$ is our $i$-th space of possible observations.
Assume that the morphisms $f_i$ are each noninvertible (having neither a left nor a right inverse). Assume also that there exists an isomorphism $h : S \rightarrow \prod_i Q_i$ which is identical with the product of morphisms (the universal arrow of the product).
What, if anything, could be said about $f_i$ and $Q_i$?