Calif Transfering ownership. Taking family off deed post-purchase

[Susan O.]

In California I purchased a duplex with a family member several months ago in 2017 end.  Our intent was to take the family member off the deed but we haven't done so yet.

In order to get the loan I used a high W-2 earning relative to purchase the home with the interest in the property shared as "tenants in Common

I wish to quickly quit-claim the deed the property into my name only and remove my family members.  Which he has agreed to do.

Would this be OK for both tax purposes and legal etc?

Also would the state property tax won't be re-assessed under prop 13 California?  We've only owned the property for 6 months