At the start of the first turn in this fight, Oichi notes your chosen foe- and that Warrior then notes that you are looking for more allies. Each in their own way, whichever of the two will challenge you to impress them before they will join you. The computer presents the method for obtaining their loyalty- defeat them within 4 turns.
The Cave map is a bit curious- with three stripes of lowered land, two bearing water and a third bearing lava, it doesn’t seem especially fiery. The two raised corners make for some difficult places to get to, or to get out of- you can usually rely on a non-flier that appears here spending an entire turn just getting down from the rocks.
Despite the obstacles, succeeding against Takatora or Rikyu is pretty easy- as long as you don’t get yourself hung up by the rocks in the corners, a decently swift charging assault will take your target out in the allotted time. This time, in addition to the screens at the end of your previous battles, you will see the Recruitment screen, showing you the information for the Warrior(s) you just defeated and can recruit, and asking you to select the ones you want. Pick your recent foe and confirm to pass this step.
The Warrior then thanks you for inviting them along, and joins your army. The game follows up by noting that you can gain the allegiance of a Warrior by defeating them in one of several ways- the typical ones are defeating them within the first 4 turns of a battle, defeating them with a super-effective move, or defeating them without any of the Pokemon your Warriors are using being hit before the defeat of the independent Warrior. Keep in mind that some free Warriors are special, and you can only attract their attention with certain other special conditions.
With that task over, Oichi compliments you on your work to increase the size of your army- and is promptly interrupted by Mitsunari, who derides her for ‘feeling special’. As the gentle Warrior reels a bit in shock, Mitsunari points out that Aurora has more than Ignis as neighbors, and this is hardly a time to be sitting around feeling special instead of actually getting some work done and conquering new areas.
Kiyomasa cuts in, pointing out that Ignis’ Warriors mostly just used Fire-type Pokemon- and while they have difficulty with water, they are exceptionally well-situated to handle grass-types- which Mitsunari points out are the specialty of the neighboring kingdom of Greenleaf, while Fontaine, your other neighbor, is mostly a locus of Water-types.
Matsunori briefly cuts in to be enraged and confused, but Kiyomasa ignores him, instead pointing out that the warlords of Greenleaf and Fontaine don’t seem to have any interest in conquering- so if you’re going to move on them, you should do so soon. Mitsunari replies that the trio don’t have time for idle talk, and have to get to the Ponigiri Shop- for the sake of their Pokemon’s energy. With that, the trio leave you be again.