It is regarding the Holy Spirit in two verses.
John 13:16 “When the Spirit of truth comes, he will guide you into all the truth, for he will not speak on his own authority, but whatever he hears he will speak, and he will declare to you the things that are to come.”
With whose authority does the Spirit speak?
Who is the Spirit of God interceding to?
Let’s assume your theology is correct that the spirit is simply the spirit of the Father. If the role of the spirit of the Father (or more simply the Father himself) is to intercede, who is the Father interceding between? Intercession as I understand it includes an intermediate who speaks on our behalf to another. Who is the Father (his spirit) interceding to for his people?
In order to understand what Christ is staying in John 13, you must know that he is speaking in parables that are not plainly understood.