| context | ${\bf C}$ … category |
| definiendum | $T$ |
| inclusion | $T:\mathrm{Ob}_{\bf C}$ |
| for all | $X:\mathrm{Ob}_{\bf C}$ |
| postulate | $\exists!i.\ i:{\bf C}[X,T]$ |
See Initial object for the definition of the dual, as well as a characterization as universal morphism.
Wikipedia: Initial and terminal objects