Universal property of total left derived functors

Observation 6.6.2. A priori, the condition on \(\bL F\) to be a right Kan extension demands that for every functor \(H\colon C[W^{-1}] \to D\), the composite \[ \Nat (H,\bL F) \xrightarrow {\gamma ^*} \Nat (H \circ \gamma , \bL F \circ \gamma ) \xrightarrow {\alpha \circ -} \Nat (H \circ \gamma , F) \] is an equivalence. But since \(\gamma ^*\) is always an equivalence by definition of localizations, this is equivalent to asking the second map to be an equivalence. We conclude that a transformation \(\alpha \colon \bL F \circ \gamma \Rightarrow F\) is a total left derived functor of \(F\) if and only if for every functor \(G\colon C \to D\) that inverts all maps in \(W\), the map \[ \alpha \circ - \colon \Nat (G, \bL F \circ \gamma ) \, \to \, \Nat (G, F) \] is an equivalence.