Yeshivas Toras Emes was founded by the Rebbe Rashab in Hebron in 1911 and later moved to Jerusalem, where it continues as a Chabad yeshivah. These letters are addressed to its administration (roughly 1947–1965). See also Rabbi Nisan Horowitz, one of its first students and mashpi'im.
What a yeshivah stands on. The Rebbe wrote that a yeshivah must stand on three pillars: Torah study, prayer and acts of kindness (IK 8536); that true education is measured not by outward appearance or superficial piety but by real inner growth (IK 1991); and that real effort in Torah study and outreach yields tangible results, while excuses are a sign of weakness (IK 8803).
Character. He taught the students to judge others favorably but be demanding with themselves, never wasting time (IK 1478), and that true breadth of mind comes through following Torah guidance (IK 2459).
Leadership and continuity. The Rebbe encouraged timely succession planning and growth in the yeshivah's leadership (IK 8639), and urged marking Yud Shevat with a farbrengen and study of the maamar "Basi LeGani" (IK 8002).
Sources: the Rebbe's letters cited above.