See the thread linked below.
Please help me understand these two sentences! ~てもらう、~てくれる | Japan Forum
The verbs attached to the -te form of a verb are called 補助動詞[hojodoushi] "subsidiary verb" in Japanese grammar. There is another category 後項動詞[koukoudoushi] "second compound verb", which is attached to the -masu stem of a verb. Plus, some adjectives also perform the same rolls.
examples of subsidiary verbs/adjectives
iru, aru, iku, kuru, shimau, oku, yaru, miru, miseru, hoshii,,,
examples of second compound verbs/adjectives
hajimeru, dasu, tsuzukeru, owaru, oeru, yamu, sugiru, au, kakeru, komu, nasaru, kaneru, yasui, nikui, zurai, gatai, gachida,,,,
Grammatically, these verbs and adjectives can be considered a kind of auxiliary verb, since they lose the original meanings and/or can attach to many verbs relatively freely. Besides these ones, there are so many compound verbs that simply combines two verbs without changing the original meanings, just like yomimachigau(yomu "to read" + machigau "to make a mistake" = "misread"), kakiwasureru(kaku "to write" + wasureru "to forget" = "to forget to write").