  We study how the category of $q$-connections depends on the choice of
coordinates. We exploit Bhatt's and Scholze's $q$-crystalline site, which is
based on a coordinate free formulation of $q$-PD structures, in order to relate
$q$-crystals and $q$-connections in the $p$-adic setting. This yields a natural
equivalence between the the categories of $q$-connections for different choices
of coordinates in the $p$-adic setting. The equivalence can be described
explicitly in terms of differential operators. In order to obtain a global
equivalence, we patch the $p$-adic differential operators to create a global
one. The process is not entirely formal, and we are only able to obtain a
global equivalence after inverting $2$.
