Universal Arrow

Given functor $F:\mathcal{C}\to \mathcal{D}$ and object $D\in \mathcal{D}$:

A universal arrow from $D$ to $F$ is:

• $(C,u:D\to F(C))$ such that
• $\forall f:D\to F(C^{\prime }),\exists !h:C\to C^{\prime }\text{ with }F(h)\circ u=f$

Encodes “initiality” or “best approximation”