I don't understand this proof, specifically the part in red, I don't understand. Please help me understand this step in the proof. Thanks!
Let Tors be the category whose objects are torsion abelian
groups; if and are torsion abelian groups, we deﬁne to
be the set of all (group) homomorphisms . Prove that direct
products exist in Tors; that is, show that given any indexed family
where each is a torsion abelian group, there exists a torsion abelian group
which serves as a direct product for this family in Tors.
Proof- Let be the torsion subgroup (that is, the subgroup of elements of ﬁnite
order) of ; here of course
is the direct product of in the category Ab. Let be
the inclusion and for each , let denote the usual projection
map; that is, . (In coordinate notation, .)
For each deﬁne by . I claim that the group
together with the maps constitute a direct product for in
Tors. Well, given a torsion group and maps for each ,
one deﬁnes as follows: given , let be the function
deﬁned by . Then clearly . Moreover,
if is any other map such that , then for any and
, , so .
I don't understand what is.