  Celestial diamonds encode the structure of global conformal multiplets in 2D
celestial CFT and offer a natural language for describing the conformally soft
sector. The operators appearing at their left and right corners give rise to
conformally soft factorization theorems, the bottom corners correspond to
conserved charges, and the top corners to conformal dressings. We show that
conformally soft charges can be expressed in terms of light ray integrals that
select modes of the appropriate conformal weights. They reside at the bottom
corners of memory diamonds, and ascend to generalized currents. We then
identify the top corners of the associated Goldstone diamonds with conformal
Faddeev-Kulish dressings and compute the sub-leading conformally soft dressings
in gauge theory and gravity which are important for finding nontrivial central
extensions. Finally, we combine these ingredients to speculate on 2D effective
descriptions for the conformally soft sector of celestial CFT.
