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Let's say I have a Nurgle sorcerer, can they cast Miasma of Pestilence on Mortarion? It works on Nurgle Daemons but Mortarion doesn't have the Faction "Daemon", it's only under "Keywords". Or does that matter?

And then could Mortarion cast the Contagion version of Miasma on himself for -2 to hit? Thanks.

First. Yes

Second. No

Faction keyword is for making the list and use of stratagems.
All powers with the same name is counted as the same power even if it’s does different things.

From the chaos daemon FAQ :

Q: For the purposes of the Psychic Focus matched play rule,
is the Miasma of Pestilence psychic power from Codex:
Chaos Daemons the same psychic power as in Codex:
Death Guard?
A: Yes. More generally, if psychic powers have the same
name, they are the same psychic power for the purposes
of Psychic Focus.

As noted, Miasma is the same power regardless of discipline.

Faction keywords get tricky, kind of.
For example:

I could build a battalion with a Death Guard DP, Poxbringer, 3x 3 Nurlglings.
They all share the NURGLE faction keyword, but as only the DP has DEATH GUARD, the detachment does not unlock DG Strats. Nor would I get any DAEMON strat because the DP does not have the DAEMON Faction keyword

But once the game starts, rules and abilities don't care about Faction keywords, just keywords in general. So my Poxbringer can cast Miasma on the DG DP.
And to add one more layer, If I select the DG DP s my WL, I DO get access to DG Traits and Relics, even though I do not technically have a DG detachment. Only having a DG WL matters for those

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