In many pages the meaning of やがて is "in the proper time", sometimes it's "eventually"… I'd like to know all the meanings.
I feel it's about observation of future but it's not completely sure to happen. Normally, it's seen in a book not in a daily conversation. Sometimes it's cool to use but might be taken weird. I think the root word for that is いつか. Casual いつか Formal 何れ、やがて
Something happening within little days/hours having passed, as well as the condition it will become. Within. Soon. Right at.
2:それにほかならない。まさに。とりもなおさず。It won't become something else (very literal here). Exactly/Certainly (intensifier). That is to say.
３ そのまま。引き続いて。In that state, Continuously.
4:４ 時を移さず。ただちに。すぐさま。 Without taking any time. Instantly. Immediately.
So, it seems to refer to something that will likely happen before a lot of time has passed. The ''In Proper time'' translation Is likely referring to the second meaning, where the ''soon'' is used in a way where the speaker feels much more certain about it, which can be used as an intensifier to what you're saying. Jisho translates it as ''In the end'' ''Finally'' and '"Eventually''.
You could say Yagate is referring to an expected predicted state that will happen within a short timeframe, and people use it to mean these 4 things.
Note that Yagate is not a word you'd use in standard daily conversation, as far as I know.