The inverse limit of $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$ is the pair $\smash {\Big(\displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright ),\webleft\{ \text{pr}_{\alpha }\webright\} _{\alpha \in I}\Big)}$ consisting of:
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The Limit. The set $\displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ defined by
\[ \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (x_{\alpha }\webright )_{\alpha \in I}\in \prod _{\alpha \in I}X_{\alpha }\ \middle |\ \begin{aligned} & \text{for each $\alpha ,\beta \in I$, if $\alpha \preceq \beta $,}\\ & \text{then we have $x_{\alpha }=f_{\alpha \beta }\webleft (x_{\beta }\webright )$} \end{aligned} \webright\} . \]
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The Cone.The collection
\[ \webleft\{ \text{pr}_{\gamma }\colon \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )\to X_{\gamma }\webright\} _{\gamma \in I} \]
of maps of sets defined as the restriction of the maps
\[ \webleft\{ \text{pr}_{\gamma }\colon \prod _{\alpha \in I}X_{\alpha }\to X_{\gamma }\webright\} _{\gamma \in I} \]
of Item 2 of Construction 2.1.2.1.2 to $\displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ and hence given by
\[ \text{pr}_{\gamma }\webleft (\webleft (x_{\alpha }\webright )_{\alpha \in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{\gamma } \]
for each $\gamma \in I$ and each $\webleft (x_{\alpha }\webright )_{\alpha \in I}\in \displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$.
We claim that $\operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ is the limit of the inverse system of sets $\webleft (X_{\alpha },f_{\alpha \beta }\webright )_{\alpha ,\beta \in I}$. First we need to check that the limit diagram defined by it commutes, i.e. that we have
\begin{align*} \webleft [f_{\alpha \beta }\circ \text{pr}_{\alpha }\webright ]\webleft (\webleft (x_{\gamma }\webright )_{\gamma \in I}\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{\alpha \beta }\webleft (\text{pr}_{\alpha }\webleft (\webleft (x_{\gamma }\webright )_{\gamma \in I}\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{\alpha \beta }\webleft (x_{\alpha }\webright )\\ & = x_{\beta }\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{pr}_{\beta }\webleft (\webleft (x_{\gamma }\webright )_{\gamma \in I}\webright ), \end{align*}
where the third equality comes from the definition of $\operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$. Next, we prove that $\operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ satisfies the universal property of an inverse limit. Suppose that we have, for each $\alpha ,\beta \in I$ with $\alpha \preceq \beta $, a diagram of the form
in $\mathsf{Sets}$. Then there indeed exists a unique map $\phi \colon L\overset {\exists !}{\to }\smash {\displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )}$ making the diagram
commute, being uniquely determined by the family of conditions
\[ \webleft\{ p_{\alpha }=\text{pr}_{\alpha }\circ \phi \webright\} _{\alpha \in I} \]
via
\[ \phi \webleft (\ell \webright )=\webleft (p_{\alpha }\webleft (\ell \webright )\webright )_{\alpha \in I} \]
for each $\ell \in L$, where we note that $\webleft (p_{\alpha }\webleft (\ell \webright )\webright )_{\alpha \in I}\in \prod _{\alpha \in I}X_{\alpha }$ indeed lies in $\operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$, as we have
\begin{align*} f_{\alpha \beta }\webleft (p_{\alpha }\webleft (\ell \webright )\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [f_{\alpha \beta }\circ p_{\alpha }\webright ]\webleft (\ell \webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}p_{\beta }\webleft (\ell \webright ) \end{align*}
for each $\beta \in I$ with $\alpha \preceq \beta $ by the commutativity of the diagram for $\webleft (L,\webleft\{ p_{\alpha }\webright\} _{\alpha \in I}\webright )$.
