## Initial morphism

### Collection

 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 $\exists_!g.\ f=U(g)\circ\phi$

### Discussion

For an elaboration, see terminal morphism, the dual concept.

#### Reference

Wikipedia: Universal property