| context | ${\bf C}$ … category |
| context | $T\in{\bf C}$ |
| definiendum | ${\bf C}/T$ |
| definition | $\mathrm{Ob}_{{\bf C}/T}:=$ all arrows $f$, such that there is an object $S\in{\bf C}$, such that $f:{\bf C}[S,T]$ |
| definition | ${\bf C}/T[f,g]:=$ all arrows $h$, such that $h\circ g = f$ |
Given a (target) object $T$ in any category ${\bf C}$, the over category ${\bf C}/T$ has objects all arrows into $T$, and arrows all forwardings of their domains (there is one for each commutative triangle).
nLab: Overcategory