The direct colimit of $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}$ is the direct colimit of $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}$ in $\mathsf{Sets}$ as in 
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1.
The Colimit. The set $\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}(X_{\alpha })$ defined by
\[ \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}(X_{\alpha })\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left.\left(\coprod _{\alpha \in I}X_{\alpha }\right)\middle /\mathord {\sim }\right., \]where $\mathord {\sim }$ is the equivalence relation on $\coprod _{\alpha \in I}X_{\alpha }$ generated by declaring $(\alpha ,x)\sim (\beta ,y)$ iff there exists some $\gamma \in I$ satisfying the following conditions:
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(a)
We have $\alpha \preceq \gamma $.
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(a)
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(b)
We have $\beta \preceq \gamma $.
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(c)
We have $f_{\alpha \gamma }(x)=f_{\beta \gamma }(y)$.
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2.
The Cocone.The collection
\[ \left\{ \mathrm{inj}_{\gamma }\colon X_{\gamma }\to \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}(X_{\alpha })\right\} _{\gamma \in I} \]of maps of sets defined by
\[ \mathrm{inj}_{\gamma }(x)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[(\gamma ,x)] \]for each $\gamma \in I$ and each $x\in X_{\gamma }$.
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1.
We have $\alpha \preceq \gamma $.
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2.
We have $\beta \preceq \gamma $.
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3.
We have $f_{\alpha \gamma }(x)=f_{\beta \gamma }(y)$.
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1Incidentally, the conditions \[ \left\{ i_{\alpha }=\phi \circ \mathrm{inj}_{\alpha }\right\} _{\alpha \in I} \]show that $\phi $ must be given by\[ \phi ([(\alpha ,x)])=(i_{\alpha }(x))_{\alpha \in I} \]for each $[(\alpha ,x)]\in \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}(X_{\alpha })$, although we would need to show that this assignment is well-defined were we to prove Construction 4.2.6.1.2 in this way. Instead, invoking Chapter 10: Conditions on Relations, of Proposition 10.6.2.1.3 gave us a way to avoid having to prove this, leading to a cleaner alternative proof.
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1.
The Prüfer Group. The Prüfer group $\mathbb {Z}(p^{\infty })$ is defined as the direct colimit
\[ \mathbb {Z}(p^{\infty })\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{n\in \mathbb {N}}(\mathbb {Z}_{/p^{n}}); \]see
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4.2.6 Direct Colimits
Let $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}\colon (I,\preceq )\to \mathsf{Top}$ be a direct system of sets.
Concretely, the direct colimit of $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}$ is the pair $\smash {\Big(\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}(X_{\alpha })}$, $\smash {\left\{ \mathrm{inj}_{\alpha }\right\} _{\alpha \in I}\Big)}$ consisting of:
Proof of Construction 4.2.6.1.2.
We will prove Construction 4.2.6.1.2 below in a bit, but first we need a lemma (which is interesting in its own right).
For each $\alpha ,\beta \in I$ and each $x\in X_{\alpha }$, if $\alpha \preceq \beta $, then we have
\[ (\alpha ,x)\sim (\beta ,f_{\alpha \beta }(x)) \]
in $\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}(X_{\alpha })$.
Proof of Lemma 4.2.6.1.3.
Taking $\gamma =\beta $, we have $f_{\alpha \gamma }=f_{\alpha \beta }$, we have $f_{\beta \gamma }=f_{\beta \beta }\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{X_{\beta }}$, and we have
\begin{align*} f_{\alpha \beta }(x) & = f_{\beta \beta }(f_{\alpha \beta }(x))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{X_{\beta }}(f_{\alpha \beta }(x)),\\ & = f_{\alpha \beta }(x).\end{align*}
As a result, since $\alpha \preceq \beta $ and $\beta \preceq \beta $ as well, Item 1a, Item 1b, and Item 1c of Construction 4.2.6.1.2 are met. Thus we have $(\alpha ,x)\sim (\beta ,f_{\alpha \beta }(x))$.
We can now prove Construction 4.2.6.1.2:
Proof of Construction 4.2.6.1.2.
We claim that $\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}(X_{\alpha })$ is the colimit of the direct system of sets $(X_{\alpha },f_{\alpha \beta })_{\alpha ,\beta \in I}$.
Commutativity of the Colimit Diagram
First, we need to check that the colimit diagram defined by $\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}(X_{\alpha })$ commutes, i.e. that we have 
\begin{align*} [\mathrm{inj}_{\beta }\circ f_{\alpha \beta }](x) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{inj}_{\beta }(f_{\alpha \beta }(x))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[(\beta ,f_{\alpha \beta }(x))]\\ & = [(\alpha ,x)]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{inj}_{\alpha }(x), \end{align*}
where we have used Lemma 4.2.6.1.3 for the third equality.
Proof of the Universal Property of the Colimit
Next, we prove that $\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}(X_{\alpha })$ as constructed in Construction 4.2.6.1.2 satisfies the universal property of a direct colimit. Suppose that we have, for each $\alpha ,\beta \in I$ with $\alpha \preceq \beta $, a diagram of the form 
\[ \left[\coprod _{\alpha \in I}i_{\alpha }\right](x)=\left[\coprod _{\alpha \in I}i_{\alpha }\right](y). \]
Proof. Indeed, if $(\alpha ,x)\sim (\beta ,y)$, then there exists some $\gamma \in I$ satisfying the following conditions:
We then have
\begin{align*} \left[\coprod _{\alpha \in I}i_{\alpha }\right](x) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}i_{\alpha }(x)\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[i_{\gamma }\circ f_{\alpha \gamma }](x)\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}i_{\gamma }(f_{\alpha \gamma }(x))\\ & = i_{\gamma }(f_{\beta \gamma }(x))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[i_{\gamma }\circ f_{\beta \gamma }](x)\\ & = i_{\beta }(y)\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left[\coprod _{\alpha \in I}i_{\alpha }\right](y). \end{align*}
This finishes the proof of the lemma. Continuing, by Chapter 10: Conditions on Relations, of Proposition 10.6.2.1.3, there then exists a map $\phi \colon \smash {\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}(X_{\alpha })}\overset {\exists !}{\to }C$ making the diagram
Here are some examples of direct colimits of sets.
