  In the paper we continue the research of Bors\'{i}k and Dobo\v{s} on
functions which allow us to introduce a metric to the product of metric spaces.
In this paper we extend their scope on broader class of spaces which usually
fail to satisfy the triangle inequality, albeit they tend to satisfy some
weaker form of this axiom. In particular, we examine the behavior of functions
preserving $b$-metric inequality. We provide analogues of the results of
Bors\'{i}k and Dobo\v{s}, adjusted to the new, broader setting. The results we
obtained are illustrated with multitude of examples. Furthermore, the
connections of newly introduced families of functions with the ones already
known from the literature are investigated.
