When creating an artwork for print, always ensure you are working in CMYK colour mode. This will allow you to work in the correct colour space for print and will increase your chances for colour accuracy.
When supplying an artwork for print, always ensure the colour mode is set to CMYK, with the profile set to Coated Fogra39. This will give you a snapshot of how the artwork will look when printed, just be aware that all monitors can display differently and that this is not a proof.
If colour accuracy is your top priority, then always get a printed proof. Every printer has a different colour gamut available to it and each has a different colour gamut depending on the substrate it is printing on. This can be even more pronounced when using different modes of print, i.e. UV print or Dye Sublimation print.
If the artwork contains any RGB elements and once converted to CMYK, there is a chance for colours to change/shift. This can happen as RGB colour space has a much larger colour Gamut available to it, particularly with bright vibrant colours, and once converted to CMYK there is a chance that these colours will dull down noticeably.
Here, I have an artwork set to RGB mode with a selection of RGB colours: