Calibrate monitor – this creates a a profile for it, with a specific filename if the calibration software allows it.

In Ps you access the currently active monitor profile from the “Proof > Custom setup” menu as “Monitor”
As you select it “Soft proofing” turns on and Ps uses the available information about the image to adjust it to show correct colors on your display.

Note that the above implies that you need to tell Ps which profile and thus color space the image data belongs in. If you don’t know, check out information on which ones are the largest and try there first… “Pro photo, Adobe RGB, …. sRGB”

As you have done this, the display should reflect the “true” colors of the image to the extent that your display allows them to be shown.

Now, having more profiled devices, say a printer, allows to select the profile for it – and emulate how a print would look. Just go select the profile in the Proof > Custom setup menu.

As the proofing profile has a smaller gamut than the current image’s gamut then you may check if there are pixels that has colors that are hard to fit within the smaller gamut:

Proof > Mark out of gamut color”

Out of gamut colors, correction (old Ps used, same technique can be used still).

Details on old photoshop, still relevant?