During the intermittent rounds of fighting in the Gaza Strip since 2008, Cairo has served as an intermediary between the Hamas regime and Jerusalem, which naturally eschew direct contact with each other. Even under the pro-Hamas rule of Muhammad Morsi, and all the more so under the anti-Islamist rule of Abdel Fattah el-Sisi, Egypt has been the sole power able to restrain the terrorist group somewhat. Eran Lerman argues that Israel benefits from having its southern neighbor in this position:
Today, the relationship [between Israel and Egypt] has reached new heights, due to their shared efforts against terror in Sinai, on one hand, and against Turkish subversion in the eastern Mediterranean, on the other hand. With a partnership in restoring calm in Gaza, and in an age of integration in the field of energy supply, there may even some change in the generally shrill anti-Israel atmosphere in the Egyptian public sphere. In this respect, the creation, in Cairo, of the Eastern Mediterranean Gas Forum (EMGF)—bringing together Egypt, Israel, Jordan, the Palestinian Authority, Cyprus, Greece, and Italy—is another step in that direction.
At the same time, the mediation is of great importance for Egypt itself. . . . A severe deterioration of the situation in Gaza, and a level of distress that may lead to pressure to throw the border open, are viewed in Cairo as a nightmare. The last thing that Egypt needs are millions more mouths to feed. Beyond that, the growing grip by Egyptian intelligence on events in Gaza can serve to force the Palestinian terror groups to cease and desist from all aid to the “Sinai Province” of Islamic State and other subversive elements in the peninsula. . . . At present, Egypt seems to have achieved an effective deterrence against further Palestinian support for terror groups in Sinai. . . .
Over time, the combination of Israeli pressures, a deterrent effect (even if limited and fragile), and intense Egyptian engagement, all help to erode the myth of the jihadist “resistance” [on which Hamas stakes its legitimacy]. The Hamas leadership’s diplomatic and military efforts of the past year are overtly designed to extract material gains. As such, they also raise—in a certain sense—questions about the movement’s ideological commitment to jihad at all costs. Thus, the very reliance upon Egypt, at times of crisis and distress, may indicate that in the regional power struggle among ideological camps, the Islamists are not quite sure that they still have the upper hand.
Read more on Jerusalem Institute for Strategy and Security: https://jiss.org.il/en/lerman-egyptian-mediation-in-gaza/