DO I NEED TO INFORM MY LOCAL COUNCIL/ GOVERNMENT OF MY SDP AWARD?

Could someone please answer this question for me. I’m currently on ESA with the SDP element. I’m also on PIP with Enhanced Disability award and Stand Mobility award. My PIP award was reviewed by telephone assessment in Early November 2020 This was after filling in the forms and posting them for the review 8months earlier in February 2020.  I was told of my PIP award entitlement early in December and they backdated the award to February which was when I applied for the reassessment. my social worker informed me that I should be entitled to the SDP element for my ESA because I’ve been awarded Enhanced Disability daily activity points where as in my previous PIP application I was only awarded points for the Mobility element of PIP.
I rang up the benefit office and they said yes I am entitled to SDP and they have backdated it to the date my PIP award was started from. I have just been on my local governments website and I looked at my register account on there to look at the information they hold about my benefit awards. I noticed that it has not been updated with my SDP benefit which was awarded from February 2020. Could someone please tell me if it is my responsibility to inform them of the SDP ? or would they already know about it but just haven’t updated the information on my account on their website? It’s just I’m worried that they will move onto UC if I inform them of SDP . Is this classed as a change in circumstances? Thanks