The initial set is the initial object of $\mathsf{Sets}$ as in 
, .
4.2.1 The Initial Set
Concretely, the initial set is the pair $\smash {\webleft (\text{Ø},\left\{ \iota _{A}\right\} _{A\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )}\webright )}$ consisting of:
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1.
The Colimit. The empty set $\text{Ø}$ of Definition 4.3.1.1.1.
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2.
The Cocone. The collection of maps
\[ \left\{ \iota _{A}\colon \text{Ø}\to A\right\} _{A\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )} \]given by the inclusion maps from $\text{Ø}$ to $A$.
Proof of Construction 4.2.1.1.2.
We claim that $\text{Ø}$ is the initial object of $\mathsf{Sets}$. Indeed, suppose we have a diagram of the form
