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Although their interactions together have been minimal compared to other characters, they have been paired together several times and appear to work well when they are.
Canonically, they both share animal-like quirks and them being classmates in 1-A allows the two to learn what they are, since they saw the tips of each other's frog and shadow quirks during their first lesson with Aizawa, along with the times they have been paired up and the Sports Festival. They appear very cooperative in fights together as well, ever since they were first paired up together in All Might's first lesson with Class 1-A, and have continued to be paired up with one another during their other team-up battle trails up to the practical section of their final exams.
During their test/fight with Ectoplasm, Tsuyu and Fumikage also appeared to have chemistry in battle, displaying elite energy and teamwork. Their quirks appear to work very well together in both evasive and combat situations. Along Fumikage using his Dark Shadow to help Tsuyu escape from their opponent when got surrounded, and to make sure that her partner doesn't get left behind Tsuyu uses her frog tong. Tsuyu believed that with Fumikage's strength they would be OK, showing that they at least slightly respect each other. Together they managed to pass their finals.
However, as stated prior, they do not interact enough to have a solid amount of canon interactions to fully judge their relationship. They do appear to be friends, although they clearly aren't as close as some other characters are. The frog/bird dynamic also helps further the possibility that they may become closer by bonding over their similarities. Like the lessons they gained during their separate internships, where they meet heroes who also have similar bird and water animal-like quirks, along Hawks teaching Fumikage how to fly and Tsuyu learning the importance of being a hero from Sirius.
This ship sailed as result as being matched in their quirk training, in addition to both having animal characteristics (Asui as a frog, and Tokoyami as a bird). This ship is also well-known in the fandom, so that's why it gets much appreciation by the fans. It is the most written for Tokoyami and the second most written for Asui. While most of the ship's fanart commonly depicts them as a princess and her prince in fairy tale AU's, since Asui's frog quirk commonly has most TokoTsuyu fans portray her as Tiana from Disney's The Princess and the Frog, or to have her as the frog while Tokoyami portrays the prince who kisses her . Along with there being fanwork that has Tokoyami's shadow being friendly with Asui or tries to get him to ask her out.
Some fan art and fics have the two meeting each other before they attended U.A.. Because Tsuyu has a friendly personality while Fumikage has a stoic-like one, fans see Tsuyu being the one who goes up to him or have Fumikage admiring her from afar. Because of Tsuyu's Frog Quirk, it has inspired fans to draw the two in the rain while sharing an umbrella. TokoTsuyu fans commonly ship it beside TodoMomo, KiriMina, IzuOcha, IidaMei, KamiJirou, BakuCamie and even BakuDeku, when they aren't rivalling TokoTsuyu with TokoShoji, TsuChako and MidoTsuyu.
On AO3, the ship has over 454 fanfics, while FanFiction.Net has 45 with only 23 placed under pairing in the My Hero Academia category. Along with TokoTsuyu having reasonable fanbases on both Tumblr and DeviantArt.
- Tsuyu's Relationship with Fumikage on the My Hero Academia Wiki
- Fumikage's Relationship with Tsuyu on the My Hero Academia Wiki