Is it because despierta is being used as an adjective here? Thank you
Because it says "sometimes I wait / for my daughter to wake up" instead of despierto which is "I wake up"
As Sherice was saying, the word despierta in this sentence is an adjective that agrees with "hija" (daughter, feminine noun -> despierta). It is not used as a verb conjugated in the present:
"Yo me despierto a las 7." (I wake up at 7.)
Here you need the ending -o for the yo-form of the verb despertar.
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