\[ \mathsf{Sets}_{*}\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{Sets}^{\mathrm{part.}} \]
between the category of pointed sets and pointed functions between them and the category of sets and partial functions between them, where:
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From Pointed Sets to Sets With Partial Functions. The equivalence
\[ \xi \colon \mathsf{Sets}_{*}\mathbin {\overset {\cong }{\rightarrow }}\mathsf{Sets}^{\mathrm{part.}} \]
sends:
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A pointed set $\webleft (X,x_{0}\webright )$ to $X$.
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A pointed function
\[ f\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright ) \]
to the partial function
\[ \xi _{f}\colon X\to Y \]
defined on $f^{-1}\webleft (Y\setminus y_{0}\webright )$ and given by
\[ \xi _{f}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (x\webright ) \]
for each $x\in f^{-1}\webleft (Y\setminus y_{0}\webright )$.
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From Sets With Partial Functions to Pointed Sets. The equivalence
\[ \xi ^{-1}\colon \mathsf{Sets}^{\mathrm{part.}}\mathbin {\overset {\cong }{\rightarrow }}\mathsf{Sets}_{*} \]
sends:
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A set $X$ is to the pointed set $\webleft (X,\star \webright )$ with $\star $ an element that is not in $X$.
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A partial function
\[ f\colon X\to Y \]
defined on $U\subset X$ to the pointed function
\[ \xi ^{-1}_{f}\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright ) \]
defined by
\[ \xi _{f}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} f\webleft (x\webright ) & \text{if $x\in U$,}\\ y_{0} & \text{otherwise.} \end{cases} \]
for each $x\in X$.