Let $\webleft (X,x_{0}\webright )$ be a pointed set.

Completeness. The category $\mathsf{Sets}_{*}$ of pointed sets and morphisms between them is complete, having in particular:
 Products, described as in Definition 3.2.3.1.1;
 Pullbacks, described as in Definition 3.2.4.1.1;
 Equalisers, described as in Definition 3.2.5.1.1.

Cocompleteness. The category $\mathsf{Sets}_{*}$ of pointed sets and morphisms between them is cocomplete, having in particular:
 Coproducts, described as in Definition 3.3.3.1.1;
 Pushouts, described as in Definition 3.3.4.1.1;
 Coequalisers, described as in Definition 3.3.5.1.1.
 Failure To Be Cartesian Closed. The category $\mathsf{Sets}_{*}$ is not Cartesian closed.^{[1]}

Morphisms From the Monoidal Unit. We have a bijection of sets^{[2]}
\[ \mathsf{Sets}_{*}\webleft (S^{0},X\webright ) \cong X, \]
natural in $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, internalising also to an isomorphism of pointed sets
\[ \textbf{Sets}_{*}\webleft (S^{0},X\webright ) \cong \webleft (X,x_{0}\webright ), \]again natural in $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.

Relation to Partial Functions. We have an equivalence of categories^{[3]}
\[ \mathsf{Sets}_{*}\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{Sets}^{\mathrm{part.}} \]
between the category of pointed sets and pointed functions between them and the category of sets and partial functions between them, where:

From Pointed Sets to Sets With Partial Functions. The equivalence
\[ \xi \colon \mathsf{Sets}_{*}\mathbin {\overset {\cong }{\rightarrow }}\mathsf{Sets}^{\mathrm{part.}} \]
sends:
 A pointed set $\webleft (X,x_{0}\webright )$ to $X$.

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 $x\in f^{1}\webleft (Y\setminus y_{0}\webright )$.

From Sets With Partial Functions to Pointed Sets. The equivalence
\[ \xi ^{1}\colon \mathsf{Sets}^{\mathrm{part.}}\mathbin {\overset {\cong }{\rightarrow }}\mathsf{Sets}_{*} \]
sends:
 A set $X$ is to the pointed set $\webleft (X,\star \webright )$ with $\star $ an element that is not in $X$.

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 $x\in X$.

From Pointed Sets to Sets With Partial Functions. The equivalence