The Torah tells in Parashat Vayigash (46:28) that Yaakov, as he made his way towards Egypt, sent his son Yehuda ahead "to point the way before him." According to one view in the Midrash, cited by Rashi, this refers to the establishment of a Torah educational facility. Understandably, Yaakov made Torah education his priority in preparing for his family's sojourn in the foreign environs and culture of Egypt.
Another Midrashic passage appears to note the novelty, or at least irony, in the fact that Yaakov chose specifically Yehuda for this task. The Midrash in Bereishit Rabba (95:1) comments that Yehuda is likened to a lion (in Bereishit 49:9), whereas Yosef is compared to cattle (Devarim 33:17). And yet, the lion and the ox, which under normal circumstances can never live together peacefully, managed to reconcile their differences and work together in harmony. "Therefore," the Midrash concludes, "'He sent Yehuda ahead of him'." Meaning, as Rav Henoch Lebovitz explains (as recorded in "Chiddushei Ha-lev"), Yaakov would not have sent Yehuda ahead to establish a yeshiva in Egypt together with Yosef, had the two been unable to iron out their differences and work together in peace and harmony.
But although this explains why Yehuda was not disqualified for this task, it does not tell us why specifically he was chosen. Rabbenu Bechayei (as well as the commentary of the Maharzu to the Midrash Rabba) claims that Yehuda was superior to his brothers in several different areas, including Torah knowledge. It was only natural, then, that Yaakov assigned him the responsibility of founding the yeshiva in Egypt.
Returning, then, to the previous Midrash, we find that although Yehuda was best suited for the job, Yaakov would have selected someone else if not for the harmony between Yosef and Yehuda. Had these two great leaders been unable to work harmoniously together, then Yaakov would not have chosen Yehuda, the most qualified of his sons to establish the yeshiva, for this job. The obvious message, as developed by Rav Lebovitz, is, as recorded by his students, "that one may not violate rules of ethics and matters of derekh eretz [etiquette and manners] even when there is a need to establish a house of study and residence for Kelal Yisrael… One should not build a place of Torah through means that oppose derekh eretz and the rules of ethics, and it is clear that if we do not abide by this [guideline], this institution will not be successful and it will not be sustained. A place of Torah built in a forbidden manner cannot possibly succeed… "
Yosef and Yehuda can join forces to build a successful religious institution only by working harmoniously together, with mutual respect and admiration, and without friction and competition.