So the basic position really amounts to saying that to have any cardinal virtue (at least in a complete and proper form), you must have all the cardinal virtues (in a complete and proper form). This makes a certain amount of intuitive sense, given that you can think of ways in which one cardinal virtue might be necessary to the completeness of another -- for instance, someone who (we suppose) has justice but not fortitude is not going to be appropriately disposed for cases in which justice requires endurance in the face of death or grave danger, and someone who has (again, we suppose) temperance but no justice would presumably have difficulty finding the right point of moderation in matters in which justice was intimately involved. In addition, of course, there are good Aristotelian reasons for thinking that all virtues whatsoever depend on prudence in one way or another. However, even so, people tend to be quite shy about accepting the unity of the virtues thesis today, so it's worth looking a bit more closely at a more developed defense. Capreolus's defense of the Thomistic version of the unity of virtues thesis, from his Defensiones Theologiae Divi Thomae Aquinatis.
Capreolus analyzes the position into basic claims that could be disputed, with each given its own basic argument.
(1) No one can possess prudence without moral virtues. You can't have, by a sort of second nature, the disposition to judge rightly in a matter to which a virtue is relevant unless you can judge rightly with respect to the particular presuppositions of particular virtues, which vary according to the virtue. We see this in cases where someone desires badly; people who desire badly have a tendency to make false assumptions in their moral reasoning about those things they desire. People who have a vice are prejudiced in the direction of the vice whenever they reason, for instance. So it follows that prudence has to grow up with the other moral virtues, so that we will be reasoning rightly in the regions of human life with which those virtues are concerned.
(2) No one can possess moral virtues without prudence. This follows in the standard Aristotelian way:
No habit that chooses according to right reason can exist without a right reason that deliberates, judges, and prescribes. But this is the way in which moral virtue is related to prudence. Therefore it cannot exist without prudence. (p. 327)
(3) The four cardinal virtues are interconnected so that whoever possesses one possesses the others. This follows, of course, from the previous two claims.
These arguments suffice for the conclusion, but as Capreolus is a scholastic commentator, he does not consider a question to have been properly understood until one can see how objections may be handled, and despite the near-universal acceptance of the unity of virtues thesis up to Capreolus's day, he is able to find some fairly formidable objections in the work of John Duns Scotus. As with many disputes between Thomists and Scotists, the dispute is not so much about the central point but about load-bearing features in the rational account of that point. Scotus, like Aquinas, does accept the unity of virtues thesis in some form, even if qualified. However, if you think about the above two arguments, it's clear that the load-bearing element in these arguments is the virtue of prudence, so that the ground for the interconnection of the cardinal virtues is prudence itself. Scotus thinks that this is more complicated than it seems, and that because of this (1) is not strictly necessarily true, and that (3) is only going to be the case if we are talking about the character of someone who is moral, simply speaking. Obviously you need all the virtues to be moral, simply speaking, and obviously to have virtue you need prudence. But Scotus has the view that there is not a single species of prudence, and you need different kinds of prudence for different virtues, so even (2) is not accepted by him in an unqualified form.
Part of the reason for the additional complications is Scotus's desire to preserve freedom of will against all possible intrusion: on his account, you can judge rightly and not will at all in accordance with it. So it is in principle possible to have the habit of judging rightly (prudence) but have no moral virtues because you simply don't make the relevant choices that your right judgments indicate. Obviously its being in principle possible does not imply that this is at all the norm; I'm not sure how common Scotus thinks it actually is, in fact. But if it's in principle possible, then (1) certainly has to be weakened and (3) qualified.
A second reason is that Scotus thinks virtues could be complete in their own field even if they are not complete with regard to what they would have to be for a completely moral character. He uses the analogy of the senses. You could have perfect sight without perfect hearing; the two kinds of sensation are different. Obviously if you lack perfect hearing, you have imperfect sensory ability, simply from that very fact, but it doesn't directly affect the act of sight. In the same way, he thinks, if you lack (say) the virtue of temperance, you lack virtue in the sense of full moral character, but this would not affect acts of (say) fortitude, for which you still could have the right disposition of second nature.
Thomists have always regarded the Scotist account as effectively breaking the account of virtues, so that virtues are barely more than habits that happen to conform to some kind of right judgment; the relationship between virtues on Scotus's account, while real, is quite limited. The acquired dispositions for the virtues aren't interconnected on their own but only in the sense that they happen to be the acquired dispositions that are consistent with right reason. The unity is posterior to the habits, one might say, rather than integral to them in the way Thomists want to argue.
The Thomistic position on free choice, of course, differs from the Scotistic position in that Scotus treats the intellect as a purely natural faculty and puts all of the freedom in the will itself, whereas Aquinas divides human freedom in matters of choice between the intellect and the will, so that both are free faculties in a limited sense and freedom of choice in the full sense arises from their interaction. Because of this, Capreolus takes the position of the Scotists to over-assimilate prudence to universal practical knowledge; prudence is in fact concerned with deliberating, judging, and prescribing in these particular circumstances here and now, and it is impossible treat the actual choice as completely separate from this judgment -- it is of the nature of the thing that how the will chooses incorporates, so to speak, the intellectual contribution. This is not to say that prudence always issues in doing the right thing, which everyone agrees is not the case with any of the virtues, but when prudence fully prescribes something as the right course, it's impossible for the will to be in a state as if this had not happened or as if this were irrelevant, and since a virtue is a habit and not a single act, it's impossible for the intellect to have the habit of prudence, and the consistent right judgment that comes with that, without the will acquiring an appropriate disposition in response.
Capreolus, of course, simply denies the analogy with the senses; the two cases are not similar. It is true that there is no strict interconnection among the senses, but this is because the sense are straightforwardly independent: sight does not subserve hearing, nor vice versa, they just contribute different things. Sensory life does not, as such, depend on any particular ordering. This is not true of moral life: moral matters are structured such that (for instance) if you are intemperate, you are going to commit injustices, and we see this all the time in people failing to do justice to each other because of lust. Moral decision-making is not, for the Thomist, a matter of doing the right thing in relatively isolated domains; rather, doing the morally right thing in one domain will often overlap with doing the morally right thing in another domain. So prudence is in fact all one and the virtues have to grow up together, so to speak, into one integral moral character. Consistently good moral decision-making is holistic, and can't be broken into lesser parts. This doesn't change the fact, noted previously, that as the virtues are being developed, the interconnections might not be fully formed; but these failures will be precisely ways in which we have failed fully to develop the virtue in question.
Now, obviously I have somewhat simplified the points in question, and equally obviously Scotists will have responses to Capreolus. But as is generally the case with scholastic dispute, as one works through the dispute one gets a clearer picture of the conclusions that are being drawn.
John Capreolus, On the Virtues, White and Cessario, trs. Catholic University Press (Washington, DC: 2001).