I think it is a mistake to correlate the analogy of the chariot and two horses exactly with the Republic's tripartition of the soul. In the Phaedrus:

All three capacities, and not only a rational part of the soul, are given an essential and positive role in striving toward the good and the beautiful, and each capacity is represented as having certain defects. Although the three capacities have some similarity to the three parts of the soul in Republic 4 - reason, appetite, and spirit - they should not simply be equated with these parts, for all three capacities share to some extent in reason and all three have desires. The charioteer represents a guiding principle in the soul, with desires of its own. The black horse represents an impulse to move in bold and disorderly fashion toward erotic objects, while the white horse represents the impulse to stand still and to resist these objects. Both horses are able to use and to follow reason and are therefore capable of being trained by the charioteer, who must also train himself to guide them without imposing excessive restraint or yielding to the impulse to move forward without any restraint. In this myth, all three capacities of the human soul share in a divine, winged element, and all three also have bestial characteristics. The black horse, then, is not innately evil but can be a force for good if he is properly trained. Without the guidance of the charioteer, the black horse moves in shameless and disorderly fashion, bending his head, stretching out his tail, taking the bit in his teeth, and dragging the chariot shamelessly forward in pursuit of an erotic object (254d6-7). After he is tamed, however, the black horse helps to move the chariot in an orderly way toward the beloved, that is, with reverence and fear (...254e9).

(Elizabeth Belfiore, 'Dancing with the Gods: The Myth of the Chariot in Plato's "Phaedrus"', The American Journal of Philology, Vol. 127, No. 2 (Summer, 2006), pp. 185-217: 190-1.)

In the Phaedrus all three capacities, represented by the charioteer and the two horses, come into agreement with one another, with the result that there is order in the soul, but this occurs only after a rough journey throughout which the soul lacks all stability. The following extract is long but the story can't be shortened:

1. The process begins when the charioteer, seeing the beloved, warms the whole soul, causing it to be filled with tickling and desire (253e5- 254a1). Although the stimulus comes first to the charioteer, all three parts of the soul have the same emotional response, characterized earlier as a boiling and tickling (251c4-5), resulting from the growth of the feathers, that affects the entire soul (251b6-7).

2. The three parts of the soul act differently in response to the same emotional stimulus. The white horse "compelled then and always by aidos, restrains himself from leaping upon the beloved" (254al-3). The black horse, however, is himself carried away by force and in turn compels (254a5, bl) the white horse and the charioteer to ap? proach the beloved and to mention the pleasures of sex (254a3-7). They at first resist (254a7-bl). Finally, however, the white horse and charioteer yield and agree to do what the black horse orders, and they approach the beloved (254b1-4).

3. When they are forced to draw near to the beloved, the charioteer sees his beauty and remembers the true beauty he saw in a previous existence (254b4-7).That is, he has the experience that was said earlier (249d4-e4) to be the madness and enthusiasm of the lover, who, seeing beauty here, is reminded of beauty there. As a result of this vision, the charioteer experiences reverence and fear (254b7-8).

4. Approach is followed by retreat. The charioteer is now compelled to pull back strongly on the reins so that both horses sit back on their haunches. The white horse obeys willingly and without resisting; the black horse obeys but much against his will. The two horses then retreat (254b8-c4).

5. The two horses react differently after the retreat. The white horse experiences shame and terror, and he waters the whole soul with sweat (254c4-5). This horse experiences not aidos, the good kind of shame that restrains him from leaping upon the beloved, but aischyne, shame at having done wrong in yielding to the black horse. When the black horse recovers from pain, he becomes angry and abusive and tries, without success, to force the others to approach the beloved again. Finally, the black horse grudgingly agrees with the others to postpone a second approach (254c5-d2). This stage of the conflict ends in a temporary truce.

6. The whole process of approach and retreat is repeated a second time (254d2-e5) and many times (pollakis, 254e6) thereafter. The black horse again compels the others to approach, pulling shamelessly to? ward the beloved, and the charioteer again pulls back on the reins. On these subsequent occasions, however, the charioteer's experience is more powerful (eti mallon, 254e1), and he pulls more strongly on the reins of the hubristic horse (eti mallon, 254e2). The white horse is not mentioned.

7. At last the black horse is tamed (tapeinotheis). He ceases from hybris and obeys the charioteer, feeling extreme fear at the sight of the beloved (254e6-8).

8. The final result of the whole process is that the whole soul of the lover follows the beloved with aidos and fear (...254e8-255al). This process results not only in the taming of the black horse but also in a permanent - or at least semi-permanent - agreement of all three parts of the soul. The charioteer is in command, and the two horses obey willingly. The whole soul now experiences both the aidos that was at first a characteristic of the white horse alone and the fear (dediduian, 254e9) that the charioteer originally experienced when he remembered true beauty at the sight of the beloved (edeise, 254b7) and that the black horse experienced while being tamed (254e8). The motion toward the beloved that the black horse once forced upon the others has now been imparted to the whole soul so that it follows the beloved in orderly fashion. In this way, the lover, having transformed disorder into order in his soul, begins to follow his god.

(Belfiore: 192-4.)