Let $N$ denote the set of positive integers that are ordered as $m \le n \iff m | n$.
Let $X_m = \mathbb{Z}/(m)$ denote the set of integers $x \mod m$.
I want to show that the direct limit of $X_i$, over $i \in N$, is $\mathbb{Q/Z}$ (using the definition here).
I have defined maps $f_{mn} : X_m \to X_n$ given by: $x \mod m \mapsto xd \mod n$, where $d$ comes from $n=md$.
I also have maps $\phi_m : X_m \to \mathbb{Q/Z}$ given by: $x \mod m \mapsto x/m$.
Then it was easy to check that $\phi_n \circ f_{mn}=\phi_m$.
Now it gets confusing for me. We have to show that if we pick another object $Y$ together with morphisms $\psi_m : X_m \to Y$ such that we have $\psi_n \circ f_{mn}=\psi_m$ then there exists a unique morphism $u \colon \mathbb{Q/Z} \to Y$ such that $u \circ \phi_m=\psi_m$ and $u \circ \phi_n=\psi_n$. I want to know how to do that?
The direct limit of the directed system $<X_m,f_{mn}>$ in the category of sets is $\sqcup X_i /\text{equivalence relation}$.
Then we define a bijection as follows \begin{align} \sqcup X_i /\text{equivalence relation} &\rightarrow \mathbb{Q/Z}\\ x_i&\rightarrow \frac{x}{i}\\ m_n&\leftarrow \frac{m}{n} \end{align} where $x_i$ is an integer in set $X_i$, $\frac{m}{n}$ are chosen such that $m<n$.