I will provide sources to show this later, but Din (דין) is the connection and transmission of intellect (מוחין) to the physical, material world.
This is expressed as the connection of Tiferet via Netzach, Hod and Yesod. ד׳י from דין is 14, an allusion to the 13 Tikunnei Dikna with the kollel. Think of וא׳ו, the miluy of Vav which alludes to Tiferet and also to the 13 Tikkunei Dikna, with the kollel is 14, ד׳י.
The Nun Sofit (ן) is Yesod d’Aba. This is in contrast to Yesod d’Nukva which is characterized by the Nun Kafufah (נ).
Din Kashe (harsh judgement) is the aspect of Leah, Mochin d’Nukva (63, ס׳ג) with the kollel (64), which is gematria Din (די׳ן).
Din Rafah (soft judgement) is the aspect of Rachel, אדנ׳י (Gematria 65, G-d's name relating to Malchut, meaning effect and expression in the physical, material world) less the Aleph, which is also Din (די׳ן).
As to the contrast of effect between Din (Din Kasheh) and Rachamim (Din Rafeh), think of the allegory of making dough and baking bread.
The effect of the leavening (שאר, which causes a decaying or fermentation effect to the flour, the concept of Gevurah and Din which also relates to the kabbalistic concept of Harisa or Haras [הריסה, הרס]) when left too long is that it sours the dough and spoils the loaf (like the example of Leah, Sag and Din Kasheh above).
The Soft Judgement, which is the aspect of Rachel, results in a beautiful and delicious loaf (like 1 held back from אדני is די׳ן, Din Rafeh.