The direct colimit of $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$ is the pair $\smash {\Big(\displaystyle \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright ),\webleft\{ \mathrm{inj}_{\alpha }\webright\} _{\alpha \in I}\Big)}$ consisting of:
-
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:
-
We have $\alpha \preceq \gamma $.
-
We have $\beta \preceq \gamma $.
-
We have $f_{\alpha \gamma }\webleft (x\webright )=f_{\beta \gamma }\webleft (y\webright )$.
-
The Cocone.The collection
\[ \webleft\{ \mathrm{inj}_{\gamma }\colon X_{\gamma }\to \operatorname*{{\displaystyle \underset {\longrightarrow }{\operatorname*{\text{colim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )\webright\} _{\gamma \in I} \]
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 $x\in X_{\gamma }$.