Objection 1. It would seem that satisfaction is not an act of justice. Because the purpose of satisfaction is that one may be reconciled to the person offended. But reconciliation, being an act of love, belongs to charity. Therefore satisfaction is an act of charity and not of justice.
Objection 2. Further, the causes of sin in us are the passions of the soul, which incline us to evil. But justice, according to the Philosopher (Ethic. v, 2,3), is not about passions, but about operations. Since therefore satisfaction aims at removing the causes of sin, as stated in the text (Sent. iv, D, 15), it seems that it is not an act of justice.
Objection 3. Further, to be careful about the future is not an act of justice but of prudence of which caution is a part. But it belongs to satisfaction, "to give no opening to the suggestions of sin" [Cf. Supplement 12, 3, Objection 1. Therefore satisfaction is not an act of justice.
On the contrary, No virtue but justice considers the notion of that which is due. But satisfaction gives due honor to God, as Anselm states (Cur Deus Homo i). Therefore satisfaction is an act of justice.
Further, no virtue save justice establishes equality between external things. But this is done by satisfaction which establishes equality between amendment and the previous offense. Therefore satisfaction is an act of justice.
I answer that, According to the Philosopher (Ethic. v, 3,4), the mean of justice is considered with regard to an equation between thing and thing according to a certain proportion. Wherefore, since the very name of satisfaction implies an equation of the kind, because the adverb "satis" [enough] denotes an equality of proportion, it is evident that satisfaction is formally an act of justice. Now the act of justice, according to the Philosopher (Ethic. v, 2,4), is either an act done by one man to another, as when a man pays another what he owes him, or an act done by one man between two others, as when a judge does justice between two men. When it is an act of justice of one man to another, the equality is set up in the agent, while when it is something done between two others, the equality is set up in the subject that has suffered an injustice. And since satisfaction expresses equality in the agent, it denotes, properly speaking, an act of justice of one man to another. Now a man may do justice to another either in actions and passions or in external things; even as one may do an injustice to another, either by taking something away, or by a hurtful action. And since to give is to use an external thing, the act of justice, in so far as it establishes equality between external things, signifies, properly speaking, a giving back: but to make satisfaction clearly points to equality between actions, although sometimes one is put for the other. Now equalization concerns only such things as are unequal, wherefore satisfaction presupposes inequality among actions, which inequality constitutes an offense; so that satisfaction regards a previous offense. But no part of justice regards a previous offense, except vindictive justice, which establishes equality indifferently, whether the patient be the same subject as the agent, as when anyone punishes himself, or whether they be distinct, as when a judge punishes another man, since vindictive justice deals with both cases. The same applies to penance, which implies equality in the agent only, since it is the penitent who holds to the penance [poenam tenet], so that penance is in a way a species of vindictive justice. This proves that satisfaction, which implies equality in the agent with respect to a previous offense, is a work of justice, as to that part which is called penance.
Reply to Objection 1. Satisfaction, as appears from what has been said, is compensation for injury inflicted. Wherefore as the injury inflicted entailed of itself an inequality of justice, and consequently an inequality opposed to friendship, so satisfaction brings back directly equality of justice, and consequently equality of friendship. And since an act is elicited by the habit to whose end it is immediately directed, but is commanded by that habit to whose end it is directed ultimately, hence satisfaction is elicited by justice but is commanded by charity.
Reply to Objection 2. Although justice is chiefly about operations, yet it is consequently about passions, in so far as they are the causes of operations. Wherefore as justice curbs anger, lest it inflict an unjust injury on another, and concupiscence from invading another's marriage right, so satisfaction removes the causes of other sins.
Reply to Objection 3. Each moral virtue shares in the act of prudence, because this virtue completes in it the conditions essential to virtue, since each moral virtue takes its mean according to the ruling of prudence, as is evident from the definition of virtue given in Ethic. ii, 6.
Source: Summa Theologica (New Advent)