Hythlodaeus describing both the office of Azem and what their stone's "special power" is
Among all the offices, the Fourteenth was most unusual. For while the rest sat in Amaurot, its holder was charged with gaining an intimate knowledge of the wider world.
In the course of his/her duty, he/she traveled the length and bredth of every land, and befriended countless folk.
He/she encountered troubles too of course. Matters which she/he could simply have referred to the Convocation. But that was not his/her way.
Nay, more often than not, he/she would call upon his/her comrades, and together resolve matters themselves.
Such is the magick sealed within that crystal - the magick to summon the starts to your side.
In time of greatest need, should you wish upon it with all your heart, it will surely answer your call.
And now it is where it belongs.
When summoning your party members to fight the WoL, this plays across the screen at the same time
Herein I commit the chronicle of the traveler. Shepherd to the stars in the dark.
Though the world be sundered and our souls set adrift, where you walk, my dearest friend, fate shall surely follow.
For yours is the Fourteenth seat - the seat of Azem.
Elidibus/Warrior of Light's reaction to you summoning your party
An invocation of eld... though not of Hydaelyn's making... What are you?
In the WoL fight, this plays across the screen after you throw off the chains in the Rift
You clutch the crystal of Azem to your brest and wish upon it with all your heart.
Your gold summoning/teleportation circle shows up again, but with Emet-Selch in the middle of it. And the WoL reacts with...
Emet-Selch does his classic finger snap, and you and the rest of your party are all pulled out of the Rift to your summoning/teleportation circle. Emet-Selch walks off and does his weird back-handed wave before disappearing again.
The phrase used to describe what you do with the crystal before Emet-Selch gets summoned/teleported is the exact same phrase Hythlodaeus uses to tell you how to use the crystal, so...
Curiously, this is how Temulun, the shaman of the Mol tribe "sees" us all the way back in Stormblood.
[You are] A singular radiance. Shimmering.
Like a jewel of the Dusk Mother, blinding in its brilliance.
Stars flicker and flock to you. Before such gathered light, even the secrets of the gods may be laid bare.
Some are fated to rise in grace and glory. Others to falter and fade, though not from memory. Both will be your strength and salvation come the end.
Cherish the stars and the light they bring you in the dark.
For you are a traveler, are you not?
When it comes to the memorial stone itself, the symbol on it is the alchemy/astrology symbol for the sun in real life. The Zodiac itself is all the constellations that the sun appears to travel through over the course of the year. It is a very appropriate symbol for the traveling Convocation seat to have.