The dogma of transubstantiation was infallibly defined by the Council of Trent:
“Certainly, the most holy Eucharist has this in common with the other Sacraments: it is the Symbol of a sacred thing, and the visible form of an invisible grace. Yet truly, there is found in it something excellent and singular. For the rest of the Sacraments first have the power of sanctifying at the time when one uses them, whereas, in the Eucharist, before being used, there is the Author of sanctity Himself.”
"And this faith has ever been in the Church of God, that, immediately after the consecration, there exists the true Body of our Lord and His true Blood, under the species of bread and wine, together with His soul and Divinity.
“But certainly, the body is under the species of bread, and the blood is under the species of wine, by the force of the words. Then the body itself is under the species of wine, and the blood is under the species of bread, and the soul is under both, by the force of that natural connection and concomitancy by which the parts of Christ our Lord – who is now resurrected from the dead to die no more – are united to one another, and also to the Divinity because of that admirable hypostatic union with His body and soul. For this reason, it is most true that as much as is contained under either species, is [contained] under both. For Christ, whole and entire, is under the species of bread, and under any part whatever of that species; again, the whole exists under the species of wine, and under its parts.”
Take note of the specifics of this infallible teaching. The substance of the bread only changes into the substance of the Body of Christ. His blood and soul, that is to say, the rest of His human nature, become present (in the same instant and in the same act) by the “natural connection and concomitancy” that all the parts of His human nature have with one another. And the Divine Nature (in the same instant and in the same act) becomes present by the “hypostatic union” that the Divine Nature has with the human nature of Christ. For the two natures are united as one Person. The same is true for the consecration of the wine, except that the substance of wine changes into the substance of Christ’s Blood, then the rest of the human nature and the Divine Nature become present in the same way as for the consecration of the bread.