Computable positive and Friedberg numberings in hyperarithmetic

We point out an existence criterion for positive computable total $\Pi^1_1$-numberings of families of subsets of a given $\Pi^1_1$-set. In particular, it is stated that the family of all $\Pi^1_1$-sets has no positive computable total $\Pi^1_1$-numberings. Also we obtain a criterion of existence for computable Friedberg $\Sigma^1_1$-numberings of families of subsets of a given $\Sigma^1_1$-set, the consequence of which is the absence of a computable Friedberg $\Sigma^1_1$-numbering of the family of all $\Sigma^1_1$-sets. Questions concerning the existence of negative computable $\Pi^1_1$- and $\Sigma^1_1$-numberings of the families mentioned are considered.