Yes. The mission of a divine person is a fitting thing, as meaning in one way the procession of origin from the sender, and as meaning a new way of existing in another because the Son is said to be sent by the Father into the world, inasmuch as He began to exist visibly in the world by taking our nature; whereas "He was" previously "in the world" (John 1:1).
Mission implies inferiority in the one sent, when it means procession from the sender as principle, by command or counsel: the one commanding is the greater, and the counsellor is the wiser. In God, however, it means only procession of origin, which is according to equality (Q42, A4; Q42, A6).
The divine person sent neither begins to exist where he did not previously exist, nor ceases to exist where He was. Hence such a mission takes place without a separation, having only distinction of origin.