Initial morphism

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.