----
> [!definition]+ Definition. ([[coslice category]])
> Let $\mathsf{C}$ be a [[category]] and $A$ an object of $\mathsf{C}$. We define a [[category]] $A / \mathsf{C}$ (or $\mathsf{C}^{A}$) for which $\text{Obj}(A / \mathsf{C})$ consists of all morphisms from $A$ to other objects of $\mathsf{C}$:
>
> ```tikz
\begin{document}
\begin{tikzpicture}
\node (A) at (0,0) {$A$};
\node (Z) at (0,-2) {$Z$};
\draw[<-] (Z) -- (A) node[midway, left] {$f$};
\end{tikzpicture}
\end{document}
>```
>
> and, given two objects $f_{1},f_{2}$ of $A / \mathsf{C}$, the set $\text{Hom}_{A / \mathsf{C}}(f_{1}, f_{2})$ is defined to be the set of all commutative diagrams
>```tikz
\begin{document}
\begin{tikzpicture}
\node (A) at (0,0) {$A$};
\node (Z1) at (-1,-1) {$Z_1$};
\node (Z2) at (1, -1) {$Z_2$};
\draw[->] (A) -- (Z1) node[midway, above left] {$f_1$};
\draw[->] (A) -- (Z2) node[midway, above right] {$f_2$};
\draw[->] (Z1) -- (Z2) node[midway, below] {$\sigma$};
\end{tikzpicture}
\end{document}
>```
> where $\sigma$ ranges over all elements of $\text{Hom}_{\mathsf{C}}(Z_{1},Z_{2})$ such that the diagram commutes ($f_{2}= \sigma f_{1}$).
>
> As in [[slice category]], composition comes from 'diagram concatenation'.
^definition
> [!generalization]
> - [[multi-object coslice category]]
^generalization
> [!basicexample]+ Example. (Pointed Sets)
> Let $\mathsf{C}=\mathsf{Set}$ and $A$ a fixed singleton $\{ * \}$. Denote $A / \mathsf{C}$ as $\mathsf{Set}^{*}$. The information of an object in $\mathsf{Set}^{*}$ — a function $f: \{ * \} \to S$ where $S$ is a set — consists of a choice of a nonempty set $S$ and an element $s \in S$. In light of this, we denote objects of $\mathsf{Set}^{*}$ as pairs $(S,s)$ where $S$ is any set and $s \in S$ any element. A morphism $(S,s) \to (T,t)$ corresponds to a set-function $\sigma:S \to T$ such that $\sigma(s)=t$.
^basic-example
----
#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```