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.