Satellites give the first derived functor

Assume the Axiom of Dependent Choice.

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories, and let $F:\mathcal{A}\to \mathcal{B}$ be additive.

If $\mathcal{A}$ has enough projectives, $F$ is right exact, and $S=(S_n)$ is any universal homological delta functor equipped with a chosen natural isomorphism $S_0\stackrel{\sim }{\to }F$, then $S_1\cong L_1^{P}F$ naturally, for every supplied projective resolution datum $P$ on all objects of $\mathcal{A}$. In this sense the first left satellite of $F$, defined as the degree-one term of a universal homological delta functor extending $F$, agrees with $L_1^{P}F$.

If $\mathcal{A}$ has enough injectives, $F$ is left exact, and $T=(T^n)$ is any universal cohomological delta functor equipped with a chosen natural isomorphism $T^0\stackrel{\sim }{\to }F$, then $T^1\cong R_I^{1}F$ naturally, for every supplied injective resolution datum $I$ on all objects of $\mathcal{A}$. This is the corresponding first right satellite agreement.