It was 1994, and when her baby daughter would fall asleep, Rowling would stroll her to a cafe that was close to her home.
In this sentence, what kind of clause is the clause "when her baby daughter would fall asleep"?
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when her baby daughter would fall asleep
is a subordinate temporal clause introduced by the subordinate conjunction when.
The scene was set in 1994, but it requires further fine tuning indicating temporal sequence introduced by time subordinator WHEN. SO,
'When the baby daughter would fall asleep...'
states the circumstances or situation of the coordinate clause in relation to which it has got its relevance:
..., Rowling would stroll her to a cafe...
The when clause here indicates the temporal visibility of time and helps situate the event of "strolling". It is an "Existential When". It is an adverb clause of time to show time relationship. When is just a subordinating conjunction and belongs to the same category as of " after, while, before,by the time, since " etc.