Is it possible to change a particle's color to the color of the face/texture part it comes out of in cycles?
I want it to be the color of the face after shading and all that other stuff
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You can make your particles materials inherit the Emitting Object's UV Coordinates to project the texture onto your particles.
You can simply unwrap the parent object (particle emitter in your case) and with the Texture Coordinates node set as in the picture, you'll be able to acces to the coordinates of the UV map of the surface point where the particle is born.
By assigning the same material to the Emitter an Particles and using the texture as a shader mask you'll be able to achieve a perfect correspondecy between the objects:
Above you can see a plane initially emitting diffusive blue particles, than some areas of the surface switch to glossy-pink shading and particles follow this change because the material is the same.