The Colimit. The set $\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ defined by
\[ \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left.\webleft(\coprod _{\alpha \in I}X_{\alpha }\webright)\middle /\mathord {\sim }\right., \]
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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We have $\alpha \preceq \gamma $.
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We have $\beta \preceq \gamma $.
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We have $f_{\alpha \gamma }\webleft (x\webright )=f_{\beta \gamma }\webleft (y\webright )$.