Normally the first medical practitioner who attends the deceased after his or her death will issue a medical death certificate.
The original of this document will be given to the funeral director that you choose.
The funeral director will then obtain your instructions to prepare an application for a death certificate from the Registry of Births Deaths and Marriages.
The original medical death certificate forms part of that application.
The original death certificate will normally be issued from the Registry to the executor, but sometimes it will be sent to the funeral director first who that will then send it to you with some certified copies.
In any case as soon as you get the death certificate you should check all the details to make sure they are correct. If there are errors you should consult your solicitor to find out if the errors will slow down your application for probate. Some errors in a death certificate render it unusable for the purpose of obtaining probate. Other errors will not prevent the Court from granting probate and therefore you are best to let the solicitor proceed with the grant using the death certificate, but you are at liberty to lodge the relevant forms at the Registry to get the register corrected.