  In this paper we study the notion of first-order part of a computational
problem, first introduced by Dzhafarov, Solomon, and Yokoyama, which captures
the "strongest computational problem with codomain $\mathbb{N}$ that is
Weihrauch reducible to $f$". This operator is very useful to prove separation
results, especially at the higher levels of the Weihrauch lattice. We explore
the first-order part in relation with several other operators already known in
the literature. We also introduce a new operator, called unbounded finite
parallelization, which plays an important role in characterizing the
first-order part of parallelizable problems. We show how the obtained results
can be used to explicitly characterize the first-order part of several known
problems.
