I agree that origin has a lot to do with it, but there's also function. I think Jesus explains how and why the Spirit of Truth is the spirit of both the Father and the Son with the following metaphor concerning his function within the human soul:
Then Jesus stood up again and continued teaching his apostles: "I am the true vine, and my Father is the husbandman. I am the vine, and you are the branches. And the Father requires of me only that you shall bear much fruit. The vine is pruned only to increase the fruitfulness of its branches. Every branch coming out of me which bears no fruit, the Father will take away. Every branch which bears fruit, the Father will cleanse that it may bear more fruit. Already are you clean through the word I have spoken, but you must continue to be clean. You must abide in me, and I in you; the branch will die if it is separated from the vine. As the branch cannot bear fruit except it abides in the vine, so neither can you yield the fruits of loving service except you abide in me. Remember: I am the real vine, and you are the living branches. He who lives in me, and I in him, will bear much fruit of the spirit and experience the supreme joy of yielding this spiritual harvest. If you will maintain this living spiritual connection with me, you will bear abundant fruit. If you abide in me and my words live in you, you will be able to commune freely with me, and then can my living spirit so infuse you that you may ask whatsoever my spirit wills and do all this with the assurance that the Father will grant us our petition. Herein is the Father glorified: that the vine has many living branches, and that every branch bears much fruit. And when the world sees these fruit-bearing branches--my friends who love one another, even as I have loved them--all men will know that you are truly my disciples. p1945:04
Jesus went on to say later:
This spirit will not speak of himself, but he will declare to you that which the Father has revealed to the Son, and he will even show you things to come; he will glorify me even as I have glorified my Father. This spirit comes forth from me, and he will reveal my truth to you. Everything which the Father has in this domain is now mine; wherefore did I say that this new teacher would take of that which is mine and reveal it to you. p1951:03
180:6.7-8 And so are you about to sorrow over my departure, but I will soon see you again, and then will your sorrow be turned into rejoicing, and there shall come to you a new revelation of the salvation of God which no man can ever take away from you. And all the worlds will be blessed in this same revelation of life in effecting the overthrow of death. Hitherto have you made all your requests in my Father's name. After you see me again, you may also ask in my name, and I will hear you. Down here I have taught you in proverbs and spoken to you in parables. I did so because you were only children in the spirit; but the time is coming when I will talk to you plainly concerning the Father and his kingdom. And I shall do this because the Father himself loves you and desires to be more fully revealed to you. Mortal man cannot see the spirit Father; therefore have I come into the world to show the Father to your creature eyes. But when you have become perfected in spirit growth, you shall then see the Father himself.
I think this might be the quote you were looking for:
20:6.8 When the mortal incarnation is finished, the Avonal of service proceeds to Paradise, is accepted by the Universal Father, returns to the local universe of assignment, and is acknowledged by the Creator Son. Thereupon the bestowal Avonal and the Creator Son send their conjoint Spirit of Truth to function in the hearts of the mortal races dwelling on the bestowal world. In the presovereignty ages of a local universe, this is the joint spirit of both Sons, implemented by the Creative Spirit. It differs somewhat from the Spirit of Truth which characterizes the local universe ages following a Michael's seventh bestowal.
20:6.9 Upon the completion of a Creator Son's final bestowal the Spirit of Truth previously sent into all Avonal-bestowal worlds of that local universe changes in nature, becoming more literally the spirit of the sovereign Michael. This phenomenon takes place concurrently with the liberation of the Spirit of Truth for service on the Michael-mortal-bestowal planet. Thereafter, each world honored by a Magisterial bestowal will receive the same spirit Comforter from the sevenfold Creator Son, in association with that Magisterial Son, which it would have received had the local universe Sovereign personally incarnated as its bestowal Son.