Some absolute properties of $A$-computable numberings

For an arbitrary set $A$ of natural numbers, we prove the following statements: every finite family of $A$-computable sets containing a least element under inclusion has an $A$-computable universal numbering; every infinite $A$-computable family of total functions has (up to $A$-equivalence) either one $A$-computable Friedberg numbering or infinitely many such numberings; every $A$-computable family of total functions which contains a limit function has no $A$-computable universal numberings, even with respect to $A$-reducibility.