It is a de fide teaching of the Church that those who die in mortal sin or in original sin alone both go to hell but suffer unequal punishments:
“The souls of those who die in mortal sin or with original sin only…immediately descend into hell, yet to be punished with different punishments". —Pope Gregory X, Second Council of Lyons, 1274
“The souls of those who depart this life in actual mortal sin, or in original sin alone, go down straightaway to hell to be punished, but with unequal pains”. —Pope Eugene IV, Council of Florence, Laetentur Caeli, 6 July 1439
When it comes to unbaptized infants, one cannot argue that because they have not committed any actual sin and are therefore “innocent,” they ought to go to heaven. A person is not saved by being “innocent,” but by having sanctifying grace, which requires baptism, whether by water, blood, or desire. Because baptism of blood and desire require an act of the will, infants can only be saved by baptism of water, which is why St. Thomas Aquinas urges mothers to have their children baptized as soon as possible, because for them there is “no other remedy.”
“It is not unjust to punish us for the sin of our first parents, because their punishment consisted in being deprived of a free gift of God…to which they had no strict right and which they willfully forfeited by their act of disobedience” (Baltimore Catechism, q. 257). No individual, even if he has not committed any personal sin, has any entitlement to a friendship with God; it is because of God’s love and mercy that Adam and Eve even had the possibility of the Beatific Vision! It is perfectly in-line with His justice if those who die in a state of original sin, with no personal sin, are still deprived of the Beatific Vision.
At the same time, though an individual who dies in a state of original sin alone cannot enter into heaven, he also cannot be punished for personal, actual sins, because he has committed none. The theory of limbo, a state of natural happiness in which unbaptized infants are deprived of the Beatific Vision but otherwise do not suffer, arose as a consequence of this theological dilemma. It is featured extensively in the writings of many saints, doctors, and fathers through the ages which have the full approval of the Church. While limbo is not explicitly mentioned in the aforementioned councils, it has become part of Catholic tradition and cannot be easily dismissed out of hand.