Universal property of determinant.

Question

Is there special name for morphism $f:X \rightarrow Y$ such that $ \ \forall g:X \rightarrow Y \ \exists!h:Y \rightarrow Y \ (g = h\circ f)$? If there is no special name, can you give examples of such morphisms? Determinant $det:GL_{n}(K)\rightarrow K^{*}$ satisfy this property, and i'm trying to find other examples.

Answer 1

I'm not aware of any name for this. As a generalization of your determinant example (modulo the single counterexample pointed out by Daniel Schepler), any abelianization $G \to G/[G, G]$ satisfies this condition in the category of groups, and more generally, if $C$ is any category and $D$ is any reflective subcategory with reflection $R : C \to D$ (left adjoint to the inclusion $D \to C$), then the unit maps $c \to R(c)$ of the adjunction satisfy this condition, by the universal property of the reflection.