baseball league: noun phrase (<em>baseball </em>functioning as an adjective, modifying <em>league</em>)
was: verb
later: adverb of time
started: verb
in: preposition
Explanation:
The given sentence has a passive voice construction, with the past simple passive verb <em>was started.</em> In that respect, the subject of the sentence is <em>The first professional baseball league</em>, and the predicate is <em>was later started in 1871</em>.
The is a determiner, first is a number, professional is an adjective, baseball is an adjective, league is a noun, was is a verb, later is an adverb, started is a verb, in is a preposition, 1871 is a number.