Yes. The Father spirates the Holy Ghost through the Son (or the Holy Ghost proceeds from the Father through the Son, which has the same meaning) because the Son receives from the Father that the Holy Ghost proceeds from Him.
If we consider in the Father and the Son the power whereby they spirate the Holy Ghost, there is no mean, for this is one and the same power.
But if we consider the persons themselves spirating, then, as the Holy Ghost proceeds both from the Father and from the Son, the Holy Ghost proceeds from the Father immediately, as from Him, and mediately, as from the Son; and thus He is said to proceed from the Father through the Son.
The same spirative power belongs to the Father and to the Son; and therefore the Holy Ghost proceeds equally from both, although sometimes He is said to proceed principally or properly from the Father, because the Son has this power from the Father.
As the begetting of the Son is co-eternal with the begetter (and hence the Father does not exist before begetting the Son), so the procession of the Holy Ghost is co-eternal with His principle. Hence, the Son was not begotten before the Holy Ghost proceeded; but each of the operations is eternal.
There is no order of power between Father and Son, but only order of 'supposita'; and hence we say that the Father spirates through the Son; and not conversely.