Sentence:  A <WSD>direct limit</WSD> has “canonical functions” which map each element of the disjoint union to its equivalence class. 
Meaning of the word marked with <WSD> tags:  A set of equivalence classes which partition the disjoint union of the members of a direct system; each equivalence class being a sort of “drainage basin” of the mappings (of the morphisms) of the direct system, if these are analogically considered as “rivers”. (If i \le k, \ j \le k in the indexing poset, then there exist f_{ik}:A_i \rightarrow A_k and f_{jk}:A_j\rightarrow A_k. If a_i \in A_i, \ a_j \in A_j such that f_{ik}(a_i) = f_{jk}(a_j) then a_i \sim a_j. If "k" = "j" then f_{jj}(a_j) = a_j, \ f_{ij}(a_i) = a_j.)