We dye polyester with pigment dye stuff. Pigment dye is a like a colored powder which we defuse into polyester . polyester FDY
For dyeing, we make a dye bath consisting of pigment , urea, emulsifiers and some other auxiliary chemicals. First the fabric passes through dye bath to pick up the content of dye bath, followed by a pressure nip roller to physically fix the particles in its place.
Then, it goes into the curing chamber which works at 150-170C. At this temperature, polyester swells and tiny pores opens up in it. And, since we need pigment dye to actually move into the fabric we need a transfer agent. Something pigment dye can ride on to go into fabric pores. At such temperature, urea decomposes to give water molecules (it absorbs from water) and those water molecules are used by pigment by pigment dye to go into fabric pores. This whole process of curing takes 5-8mins in continuous operation. And, after curing ,the fabric comes out hot and cools down in room temperature . Then, it is washed once or twice to remove excess dye and good to go. Washing is sometimes perform simultaneously after curing.

The fixation /curing of dye into fabric is a combination of riding on water and capillary diffusion into fabric . And that is why polyester fabrics never fades. Because dye is actually inside the fabric. And since polyester is oil/petroleum based fabric , we can use gasoline to remove tough stains from the fabric . And never ever use heating method to remove stains. (talco powder plus heated iron to remove grease stains).

And, if we want to dye small quantity of polyester fabric or if it is knitted fabric , delicate to dye, then, we use batch dyeing machine called Jigs. It is a high temperature pressure horizontal cylindrical reactor , where a roll of fabric is loaded like tape in cassettes and it rotates for one hour to 90min under high pressure and temperature to physically force dye into fabric.

Why we use pigment dyes (sometimes called direct dyes) for polyester? Because polyester is very stable and doesn't easily react with dye stuff, so we have to force dye into it. There are some differences in pigment and direct dyes(typically used for cotton fabrics). All pigment dyes are direct dyes but not all direct dyes are pigment. Because, especially for polyester, which has very compact structure, we need a dye with small particle size so that it can easily go into fabric.

And, due the same reason of small particle size, we cannot use pigment dye on Cotton. Because cotton has big pores and dye particles can easily go and. Come out leading to excessive color bleeding.

If you have cotton polyester fabric, it gets interesting. You first dye either cotton or polyester then the other one in step by step method. Sequence sometimes depends on type color (light color is dyed first) , cotton dyes type etc.

And, for printing on polyester fabric, same dye but different chemical mixture which is thick/viscous enough to retain the Design pattern until curing.