Print colors different from Lightroom and photoshop and Monitor

Thank you Conrad for your expert reply. For those who want more, should have a look at Conrad's brilliant book: https://www.peachpit.com/store/adobe-photoshop-classroom-in-a-book-2024-release-9780138262525

When I export the image as a jpeg it looks the same as it does in Lightroom but if I put that same photo on my iphone it's warmer.

By @hifocused

Are TrueTone or NightShift active on the iPhone? Also, assuming all other factors are equal, whether any two screens match partly depends on their current calibration and applied display profile. For example, if one display is set to a white point of 6500K and the other is set to 5000K, well of course the 5000K display is going to look warmer. I don’t think there is any control over this on an iPhone, but the Apple Studio Display supports Reference Mode presets. Open System Settings, click Displays, and look at the Preset menu. Which Reference Mode preset name is the Studio Display set to? If the specs of that Reference Mode are in any way different than how the iPhone screen currently behaves, then the same image isn’t going to look the same on both the display and the phone.

I don’t have a Studio Display, but my guess is that its default Reference Mode preset of Apple Display (P3-600 nits) is probably supposed to be a good match to an iPhone. However, keep in mind that this might never result in an exact match, because if any characteristic of the display is different between the Studio Display and iPhone (and there probably are differences), the match will only be “as close as possible” but still different. For example, you can have two cars and tell both to go 100 miles per hour, but that doesn't mean they will do it the exact same way. One car might not be physically capable of going that fast and max out at a lower speed, another car might be able to reach 100MPH but it takes longer than the other car, and so on. In the same way, you can expect two well-adjusted displays to look very close, but maybe never exactly. (Some displays that cost more than the Studio Display do provide fine-tuning controls to match two of those.)

If I put that same photo in photoshop and print from there it's wamer and less contrast also if I export it from PS as a jpeg PS changes the colors as well just on the export.   What is the best way to have consistant colors between all plate forms from LR, to PS to print?  Any help would be great thank you!

By @hifocused

Adobe photo apps use a color management system to achieve color consistency through color profiles, so everything depends on if the hardware and applications are set up properly for the color management system. (This is a whole area of study that a lot of people find confusing.) We already talked about hardware, so this part is about the applications. The colors should be consistent if:

• The display is calibrated or at least profiled. The Studio Display should be pretty good out of the box so this part might be taken care of…if the current Reference Mode preset is appropriate for the work you are doing. 
• The color settings in the applications are appropriate. In Lightroom Classic, this is automatic. In Photoshop, this is set in Edit > Color Settings…what is the name of the Settings preset selected at the top? 
• The color settings on export are appropriate. If exporting from Lightroom Classic and opening in Photoshop, you set this in the File > Export dialog box, File Settings section, Color Space. But the thing is, Lightroom Classic always embeds a color profile, so Photoshop should be using that profile to make it look the same in Photoshop. So if you are seeing them not match, it may be a misconfiguration in Photoshop, like maybe it’s set up to ignore embedded profiles. 

One more thing: Displays can reproduce a much wider range of tones (contrast) and colors than ink and paper can, so it’s normal for a print to not be able to fully match what you see on the screen. Both Photoshop and Lightroom Classic have a feature called soft-proofing that lets you use a printer color profile to simulate what your image will look like under a specific combination of printer, ink, and paper (since all three affect what the print looks like). So compared to the normal full-range view of an image, soft-proofing gives you a better sense of how it’s actually going to print, what the print limitations are. For example, high quality glossy paper in a pro-level color printer tends to print images that look a lot closer to the screen than a home/office printer with cheap matte paper in it, and both of those prints will have more contrast and color than printing on newsprint. Soft-proofing can show you the difference between the image on screen, and how that image is going to affected by printing it in any of those ways. The way you use the soft-proof view is to adjust the image while viewing it under the limitations of the target output.