| context | $X:\mathrm{Ob}_{\bf C}$ |
| context | $U$ in ${\bf D}\longrightarrow{\bf C}$ |
| definiendum | $\langle A,\phi\rangle$ in $\mathrm{it}$ |
| inclusion | $A:\mathrm{Ob}_{\bf D}$ |
| inclusion | $\phi:{\bf C}[X,U(A)]$ |
| for all | $B:\mathrm{Ob}_{\bf D}$ |
| for all | $f:{\bf C}[X,U(B)]$ |
| range | $g:{\bf D}[A,B]$ |
| postulate | $\exists_!g.\ f=U(g)\circ\phi$ |
For an elaboration, see terminal morphism, the dual concept.
Wikipedia: Universal property