The Stacks project

Definition 13.14.2. Assumptions and notation as in Situation 13.14.1. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$.

1. we say the right derived functor $RF$ is defined at $X$ if the ind-object

   $$(X/S) \longrightarrow \mathcal{D}', \quad (s : X \to X') \longmapsto F(X')$$

   is essentially constant¹; in this case the value $Y$ in $\mathcal{D}'$ is called the value of $RF$ at $X$.

2. we say the left derived functor $LF$ is defined at $X$ if the pro-object

   $$(S/X) \longrightarrow \mathcal{D}', \quad (s: X' \to X) \longmapsto F(X')$$

   is essentially constant; in this case the value $Y$ in $\mathcal{D}'$ is called the value of $LF$ at $X$.

By abuse of notation we often denote the values simply $RF(X)$ or $LF(X)$.