Yom Kippur 5778

The Mishnah in Yoma (8:9) notes that Yom Kippur only atones for transgressions between Man and G-d, but does not atone for transgressions between and Man and his friend until he appeases his friend. Rabbi Elazar ben Azariah brings a verse from the Yom Kippur service to show this: "You shall be purified from yours sins before G-d" (Leviticus 16:30). Before G-d, but not friends.
Rabbi Joseph B. Soloveichik asks on the need to bring a verse for this lesson of appeasement. The Mishnah in Baba Kamma (8:7) teaches that when one injures his fellow Jew, the former must pay numerous fees such as for damages, embarrassment, and doctor bills. Even when payment is complete, the person is not forgiven until he seeks mechila (pardoning) from his friend. We learn this from Abraham and Avimelech. G-d told Avimelech to ask Abraham for forgiveness after taking Sarah away from him; this would lead to G-d healing Avimelech. Based on this Mishnah, we know that there is a principle to ask for forgiveness from your friend when you do something wrong. Why does Rabbi Elazar need to bring a second verse?
The Rav answers that Rabbi Elazar recognizes that there is a general rule of asking forgiveness which applies throughout the year. Here, however, he is only addressing the atonement of Yom Kippur. With Yom Kippur, it's not just about asking for forgiveness. It's about actually appeasing your friend and reverting back to the friendship that existed before any wrongdoing. You are truly making up with your friend and your relationship with him.
This is crucial to accomplish because the atonement of Yom Kippur is communal. We want to include everyone within the community without any trace of separation or division among us.
The Rav goes onto add that some rabbis from the mussar movement say that Yom Kippur won't atone for transgressions between Man and G-d if one doesn't appease his friend to resolve his transgressions between him and his friend first. Why is this?
The same Mishnah above ends with Rabbi Akiva calling G-d the mikveh of Israel. G-d is Who purifies us. However, a mikveh only works in totality. You can't purify just one part of your body; it must be your whole body. So too here, where you must be completely free of transgressions from every aspect and not just in one category.