A Universal morphism from X to a functor F is a pair $(A, u : X → F A)$ such that there exists a unique map $A → A'$ such that the following diagram commutes
In other words, $(A, u)$ it is the initial object in the slice category $\mathcal{C}/F$.
Is the UMP for $u$ always arising from an adjunction? In particular, is it true that I can always define a $G$ to be the left adjoint of $F$ such that $GX = A$?
If not, is there a UMP for a particular $u$ that does not arise from an adjunction?