Objection 1. It would seem that one man cannot fulfill satisfactory punishment for another. Because merit is requisite for satisfaction. Now one man cannot merit or demerit for another, since it is written (Psalm 61:12): "Thou wilt render to every man according to his works." Therefore one man cannot make satisfaction for another.
Objection 2. Further, satisfaction is condivided with contrition and confession. But one man cannot be contrite or confess for another. Neither therefore can one make satisfaction for another.
Objection 3. Further, by praying for another one merits also for oneself. If therefore a man can make satisfaction for another, he satisfies for himself by satisfying for another, so that if a man satisfy for another he need not make satisfaction for his own sins.
Objection 4. Further, if one can satisfy for another, as soon as he takes the debt of punishment on himself, this other is freed from his debt. Therefore the latter will go straight to heaven, if he die after the whole of his debt of punishment has been taken up by another; else, if he be punished all the same, a double punishment will be paid for the same sin, viz. by him who has begun to make satisfaction, and by him who is punished in Purgatory.
On the contrary, It is written (Galatians 6:2): "Bear ye one another's burdens." Therefore it seems that one can bear the burden of punishment laid upon another.
Further, charity avails more before God than before man. Now before man, one can pay another's debt for love of him. Much more, therefore, can this be done before the judgment seat of God.
I answer that, Satisfactory punishment has a twofold purpose, viz. to pay the debt, and to serve as a remedy for the avoidance of sin. Accordingly, as a remedy against future sin, the satisfaction of one does not profit another, for the flesh of one man is not tamed by another's fast; nor does one man acquire the habit of well-doing, through the actions of another, except accidentally, in so far as a man, by his good actions, may merit an increase of grace for another, since grace is the most efficacious remedy for the avoidance of sin. But this is by way of merit rather than of satisfaction. on the other hand, as regards the payment of the debt, one man can satisfy for another, provided he be in a state of charity, so that his works may avail for satisfaction. Nor is it necessary that he who satisfies for another should undergo a greater punishment than the principal would have to undergo (as some maintain, who argue that a man profits more by his own punishment than by another's), because punishment derives its power of satisfaction chiefly from charity whereby man bears it. And since greater charity is evidenced by a man satisfying for another than for himself, less punishment is required of him who satisfies for another, than of the principal: wherefore we read in the Lives of the Fathers (v, 5) of one who for love of his brother did penance for a sin which his brother had not committed, and that on account of his charity his brother was released from a sin which he had committed. Nor is it necessary that the one for whom satisfaction is made should be unable to make satisfaction himself, for even if he were able, he would be released from his debt when the other satisfied in his stead. But this is necessary in so far as the satisfactory punishment is medicinal: so that a man is not to be allowed to do penance for another, unless there be evidence of some defect in the penitent, either bodily, so that he is unable to bear it, or spiritual, so that he is not ready to undergo it.
Reply to Objection 1. The essential reward is bestowed on a man according to his disposition, because the fulness of the sight of God will be according to the capacity of those who see Him. Wherefore just as one man is not disposed thereto by another's act, so one man does not merit the essential reward for another, unless his merit has infinite efficacy, as the merit of Christ, whereby children come to eternal life through Baptism. On the other hand, the temporal punishment due to sin after the guilt has been forgiven is not measured according to the disposition of the man to whom it is due, since sometimes the better man owes a greater debt of punishment. Consequently one man can merit for another as regards release from punishment, and one man's act becomes another's, by means of charity whereby we are "all one in Christ" (Galatians 3:28).
Reply to Objection 2. Contrition is ordained against the guilt which affects a man's disposition to goodness or malice, so that one man is not freed from guilt by another's contrition. In like manner by confession a man submits to the sacraments of the Church: nor can one man receive a sacrament instead of another, since in a sacrament grace is given to the recipient, not to another. Consequently there is no comparison between satisfaction and contrition and confession.
Reply to Objection 3. In the payment of the debt we consider the measure of the punishment, whereas in merit we regard the root which is charity: wherefore he that, through charity, merits for another, at least congruously, merits more for himself; yet he that satisfies for another does not also satisfy for himself, because the measure of the punishment does not suffice for the sins of both, although by satisfying for another he merits something greater than the release from punishment, viz. eternal life.
Reply to Objection 4. If this man bound himself to undergo a certain punishment, he would not be released from the debt before paying it: wherefore he himself will suffer the punishment, as long as the other makes satisfaction for him: and if he do not this, then both are debtors in respect of fulfilling this punishment, one for the sin committed, the other for his omission, so that it does not follow that one sin is twice punished.
Source: Summa Theologica (New Advent)