The Stacks project

Definition 12.12.3. Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $F = (F^ n, \delta _ F)$ be a $\delta $-functor from $\mathcal{A}$ to $\mathcal{B}$. We say $F$ is a universal $\delta $-functor if and only if for every $\delta $-functor $G = (G^ n, \delta _ G)$ and any morphism of functors $t : F^0 \to G^0$ there exists a unique morphism of $\delta $-functors $\{ t^ n\} _{n \geq 0} : F \to G$ such that $t = t^0$.