There is an interesting early Christian document called the Διδαχή, translated into English as "The Teaching." The word seems to be classical, not just Koine. Is this some kind of more general "that which" or "activity involving" construction? If so, is it constructed from a verb like διδάσκω, or is it some kind of noun construction like English "youtubing?"
διδαχή is indeed built on διδάσκω, though without the inchoative infix -σκ-: the root is διδαχ-, as can be seen in the aorist ἐδίδαξα, and -σκ- is one of those infixes that only show up in the present. The reduplication, in this case, indicates a causative (the original verb is the unattested-in-the-present *δάω 'to learn', though it does show up in the aorist as ἐδάην), and for semantic reasons that is preserved (and, indeed, piled onto in the perfect: δεδίδαχα).
Greek has a lot of suffixes that form nouns out of verbs, but the simplest one is just the first-declension ending ‑η. This was a mechanism that was already available in Proto-Indo-European (when the ending was still *‑eh₂) where it could be added to the o-grade of verb roots (as in the Greek descendants τρέφω 'to nourish' → τροφή 'nourishment'; διαφέρω 'to differ' → διαφορά 'difference' (ᾱ instead of η after ι, ε, or ρ in Attic); λείβω 'to pour' → λοιβή 'libation'), the e-grade (στέγω 'to cover' → στέγη 'roof, shelter' (note the accent)), or the zero-grade (ἄρχω 'to begin; to rule' → ἀρχή 'beginning; rule'; φεύγω 'to flee' → φυγή 'flight'), but the construction remained productive well into historical times. διδαχή is formally built on a zero-grade (root *dens-, zero-grade *dn̥s-), but really it's post-PIE, when those distinctions no longer make much synchronic sense.
Though this is often casually described as creating verbal nouns that's misleading: the meaning, for a verb X, isn't the act of doing X so much as an instance of Xing, as you can see. διδαχή is a teaching, not the act of teaching.
Very interesting, +1 for you– MPWFeb 18, 2021 at 4:24
Great answer. I would modify "instance of Xing" to a slightly more pedantic "denoting a state of having executed/completed/established an/the act of X". Otherwise 'instance' might also include things like «δίδαγμα» (which refers more to the outcome of the act, "that which has been taught", rather than the fact that such an act has been executed to completion in the first place). I view it similarly in this case to 'advise -> advisement', as opposed to 'advice'. (in fact, I suspect the -ment suffix in English plays a similar role to what you describe?) Feb 18, 2021 at 10:23