Holy Spirit is the Spirit of the Father because He (Spirit) issues from Him (Father) (John 15:26), and is also the Spirit of the Son, because He (Spirit) is dispensed by Him (the Son) to humans (John 20:22) and because belief in Him (Son) is the condition for Holy Spirit to gush forth in us as streams of "living water" (John 7::38).
Thus, yes, according to apostles and according to common sense based on their sayings and writings, the Holy Spirit is the Spirit of the Father who teaches us to worship the Father's Son, called Jesus Christ after the Incarnation, alongside with the Father as Lord and God (1 Cor. 12:3).