  A coprime commutator in a profinite group $G$ is an element of the form
$[x,y]$, where $x$ and $y$ have coprime order and an anti-coprime commutator is
a commutator $[x,y]$ such that the orders of $x$ and $y$ are divisible by the
same primes. In the present paper we establish that a profinite group $G$ is
finite-by-pronilpotent if the cardinality of the set of coprime commutators in
$G$ is less than $2^{\aleph_0}$. Moreover, a profinite group $G$ has finite
commutator subgroup $G'$ if the cardinality of the set of anti-coprime
commutators in $G$ is less than $2^{\aleph_0}$.
