The direct colimit of $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$ is the direct colimit of $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$ in $\mathsf{Sets}$ as in 
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1.
The Colimit. The set $\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ defined by
\[ \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left.\left(\coprod _{\alpha \in I}X_{\alpha }\right)\middle /\mathord {\sim }\right., \]where $\mathord {\sim }$ is the equivalence relation on $\coprod _{\alpha \in I}X_{\alpha }$ generated by declaring $\webleft (\alpha ,x\webright )\sim \webleft (\beta ,y\webright )$ iff there exists some $\gamma \in I$ satisfying the following conditions:
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The Cocone.The collection
\[ \left\{ \mathrm{inj}_{\gamma }\colon X_{\gamma }\to \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )\right\} _{\gamma \in I} \]of maps of sets defined by
\[ \mathrm{inj}_{\gamma }\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (\gamma ,x\webright )\webright ] \]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 $f_{\alpha \gamma }\webleft (x\webright )=f_{\beta \gamma }\webleft (y\webright )$.
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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\[ \phi \webleft (\webleft [\webleft (\alpha ,x\webright )\webright ]\webright )=\webleft (i_{\alpha }\webleft (x\webright )\webright )_{\alpha \in I} \]for each $\webleft [\webleft (\alpha ,x\webright )\webright ]\in \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$, 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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The Prüfer Group. The Prüfer group $\mathbb {Z}\webleft (p^{\infty }\webright )$ is defined as the direct colimit
\[ \mathbb {Z}\webleft (p^{\infty }\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{n\in \mathbb {N}}\webleft (\mathbb {Z}_{/p^{n}}\webright ); \]see
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4.2.6 Direct Colimits
Let $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}\colon \webleft (I,\preceq \webright )\to \mathsf{Top}$ be a direct system of sets.
Concretely, the direct colimit of $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$ is the pair $\smash {\Big(\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )}$, $\smash {\left\{ \mathrm{inj}_{\alpha }\right\} _{\alpha \in I}\Big)}$ 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
\[ \webleft (\alpha ,x\webright )\sim \webleft (\beta ,f_{\alpha \beta }\webleft (x\webright )\webright ) \]
in $\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$.
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 }\webleft (x\webright ) & = f_{\beta \beta }\webleft (f_{\alpha \beta }\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{id}}_{X_{\beta }}\webleft (f_{\alpha \beta }\webleft (x\webright )\webright ),\\ & = f_{\alpha \beta }\webleft (x\webright ).\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 $\webleft (\alpha ,x\webright )\sim \webleft (\beta ,f_{\alpha \beta }\webleft (x\webright )\webright )$.
We can now prove Construction 4.2.6.1.2:
Proof of Construction 4.2.6.1.2.
We claim that $\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ is the colimit of the direct system of sets $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$.
Commutativity of the Colimit Diagram
First, we need to check that the colimit diagram defined by $\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ commutes, i.e. that we have 
\begin{align*} \webleft [\mathrm{inj}_{\beta }\circ f_{\alpha \beta }\webright ]\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{inj}_{\beta }\webleft (f_{\alpha \beta }\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (\beta ,f_{\alpha \beta }\webleft (x\webright )\webright )\webright ]\\ & = \webleft [\webleft (\alpha ,x\webright )\webright ]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{inj}_{\alpha }\webleft (x\webright ), \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 $\operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ 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 
\[ \left[\coprod _{\alpha \in I}i_{\alpha }\right]\webleft (x\webright )=\left[\coprod _{\alpha \in I}i_{\alpha }\right]\webleft (y\webright ). \]
Proof. Indeed, if $\webleft (\alpha ,x\webright )\sim \webleft (\beta ,y\webright )$, then there exists some $\gamma \in I$ satisfying the following conditions:
We then have
\begin{align*} \left[\coprod _{\alpha \in I}i_{\alpha }\right]\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}i_{\alpha }\webleft (x\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [i_{\gamma }\circ f_{\alpha \gamma }\webright ]\webleft (x\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}i_{\gamma }\webleft (f_{\alpha \gamma }\webleft (x\webright )\webright )\\ & = i_{\gamma }\webleft (f_{\beta \gamma }\webleft (x\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [i_{\gamma }\circ f_{\beta \gamma }\webright ]\webleft (x\webright )\\ & = i_{\beta }\webleft (y\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left[\coprod _{\alpha \in I}i_{\alpha }\right]\webleft (y\webright ). \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 $\phi \colon \smash {\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\operatorname {\mathrm{colim}}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )}\overset {\exists !}{\to }C$ making the diagram
Here are some examples of direct colimits of sets.
