===== Initial morphism =====
==== Collection ====
| @#55CCEE: context     | @#55CCEE: $X:\mathrm{Ob}_{\bf C}$ |
| @#55CCEE: context     | @#55CCEE: $U$ in ${\bf D}\longrightarrow{\bf C}$ |
| @#FFBB00: definiendum | @#FFBB00: $\langle A,\phi\rangle$ in $\mathrm{it}$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $A:\mathrm{Ob}_{\bf D}$ |
| @#AAFFAA: inclusion   | @#AAFFAA: $\phi:{\bf C}[X,U(A)]$ |
| @#FFFDDD: for all     | @#FFFDDD: $B:\mathrm{Ob}_{\bf D}$ |
| @#FFFDDD: for all     | @#FFFDDD: $f:{\bf C}[X,U(B)]$ |
| @#DDDDDD: range       | @#DDDDDD: $g:{\bf D}[A,B]$ | 
| @#55EE55: postulate   | @#55EE55: $\exists_!g.\ f=U(g)\circ\phi$ |

==== Discussion ====
For an elaboration, see [[terminal morphism]], the dual concept.

=== Reference ===
Wikipedia: [[http://en.wikipedia.org/wiki/Universal_property|Universal property]]
==== Parents ====
=== Context ===
[[Functor]]
=== Requirements ===
[[Product type]]