Slice category

Remarks

It is the comma category $(1_{\mathcal{C}} \downarrow X)$, where $X$ also denotes the functor from the one-morphism category that picks $X$. Dually, the coslice category $X/\mathcal{C}$ has as objects the morphisms $X \to A$, and morphisms $f$ with $f \circ h = h'$. Example: in the category of sets, an object of $\mathbf{Set}/\set{0, 1}$ is a set $A$ with a map $h \colon A \to \set{0, 1}$, that is, a set split into two labelled parts. Write $e$ for the identity map of $\set{0, 1}$; then $e$ is a morphism from $(\set{0, 1}, e)$ to itself, since $e \circ e = e$. Non-example: let $s \colon \set{0, 1} \to \set{0, 1}$ swap $0$ and $1$. The identity map $e$ is not a morphism from $(\set{0, 1}, e)$ to $(\set{0, 1}, s)$, since $s \circ e = s \ne e$. The object $(X, 1_X)$ is a terminal object of $\mathcal{C}/X$.