Definition 13.15.2. Assumptions and notation as in Situation 13.15.1. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$.

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 constant

^{1}; in this case the value $Y$ in $\mathcal{D}'$ is called the*value of $RF$ at $X$*.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 $LF(X)$.

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