Abstract <p>The paper discusses two approaches to defining the computability of numberings of families of total functions. We consider both the classical definition of computable numbering of a family of computable functions, according to which the number of a function in this numbering effectively provides its Gödel number, and, expanding the previous one, a definition based on the uniform application of the concept of the left-c.e. element of Baire space. The main question studied in the paper is the possibility of generating all computable numberings of a family by the closure with respect to the reducibility of infinite direct sums of uniform sequences of its single-valued, positive, and minimal numberings.</p>