WARNING: The following contains spoilers for My Hero Academia Season 5, Episode 7, "Match 3," now streaming on Crunchyroll, Funimation and Hulu.
Since My Hero Academia began, the students of UA have been tasked with the responsibility of being the next generation of heroes. For some of these characters, that means directly taking on the legacy of those who've come before them, as seen with Midoriya, Todoroki, Shinso, Tokoyami and Iida in Season 5. These students, whether they like it or not, continue in the footsteps of the professional heroes they have studied under and worked for. Season 5 hammers home how important it is for this generation of heroes to carry on these heroic legacies, especially after All Might was forced into retirement.
All Might and Midoriya
The most prominent legacy hero is All Might, as his Quirk by design requires the hero to pass it down to a successor. All Might continues the legacy of his master, Nana Shimura, who was carrying on the legacy of the hero before her and so on.
Now it's Midoriya's turn to play this role, mentoring under All Might before his time at UA even began. Now with All Might in retirement, it's all the more important for Midoriya to carry on this legacy, especially with the public's trust in heroes wavering and with the threat of the League of Villains still out there.
Endeavor and Todoroki
While Midoriya is more than willing to become All Might's successor, Todoroki was forced to take on Endeavor's legacy by the new Number One Hero himself. Since he was little, Endeavor pushed Todoroki to be his successor, one that would eventually surpass All Might. This selfish desire led to Endeavor abusing and neglecting his wife and children, including Todoroki.
In Season 5, Episode 7, "Match 3," viewers see how physically and emotionally tolling this weight has been, as a flashback shows a younger Todoroki exhausted, crying and bruised after being forced to perfect a move his older brother, Toya, could not even handle. While this treatment has strained their relationship, Todoroki still wants to be a hero of his own, and he knows how working with his father can be beneficial, interning under him in Season 2. He also understands that it is on him to uphold the honor of the Number One Hero, especially when the public has not fully embraced Endeavor as a symbol of peace like they did All Might.
Eraserhead and Shinso
While Eraserhead technically is passing his legacy onto the students of Class 1-A, there is now a more direct student taking on his role for the next generation. Shinso debuted during the Sports Festival Arc, hoping his performance in the tournament would allow him to enter the Hero Course, as his Quirk, while incredibly powerful, was not well suited for the entrance exam. Like Aizawa's Erasure Quirk, Shinso's Brainwashing Quirk makes him a force to be reckoned with; however, both need to develop their physical strength, combat skills, speed and more to stand on equal footing with students and pro heroes whose Quirks give them a more physical advantage.
Eraserhead clearly sees the potential in Shinso, mentoring him, showing him how to use his capture weapon and giving him a chance to try out for the Hero Course. With Shinso learning directly from Aizawa, a hero with similar strengths and struggles, it's likely he'll take on Eraserhead's role for this generation.
Hawks and Tokoyami
Where Shinso and Aizawa are fairly similar in terms of personality, Hawks and Tokoyami are vastly different, with the former giving off a more chill presence while the latter is often serious. However, Hawks chose Tokoyami to be his intern for several reasons. He primarily chose Tokoyami so he could talk to a student from Class 1-A about the USJ attack; however, he also picked him because they're "birds of a feather," and he thought Tokoyami was wasting his potential, eventually teaching the UA student how to fly using Dark Shadow.
Even though Hawks is not interested in training the next generation, Tokoyami is an exception, as "the ones who can fly, should fly." This revelation changes Tokoyami's attitude about Hawks, and like Todoroki, he knows it's up to him to honor the Number Two Hero's legacy.
Ingenium and Iida
While the aforementioned students worked directly with their heroes at some point, Iida and Ingenium are different. Ingenium is Iida's big brother, and he's been an inspiration for the aspiring hero since he was a little kid. In fact, the Iida family itself is well known in the hero world, with each generation carrying on the past's legacy, as well as their Quirk.
Tragically, Ingenium fell Victim to Stain, thankfully surviving, but his time as a hero has come to an end. This hit Iida hard, with the student defying everything he stood for in an attempt to exact revenge on The Hero Killer. While he's moved beyond this, what happened to his brother is still impacting him, as Ingenium asks Iida directly to carry on his mantle as the new Ingenium. It's also in "Match 3," that audiences learn the Iida family has a painful way of upgrading their Quirk, with Ingenium finally passing down this knowledge to Iida so he can better carry on his legacy.