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

Parents

Context

Functor

Requirements

Product type