**Proof.**
The map is constructed as follows. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $K$. Choose a quasi-isomorpism $\mathcal{I}^\bullet \to \mathcal{J}^\bullet $ where $\mathcal{J}^\bullet $ is a K-injective complex of abelian groups. Then the map in (1) is given by $\Gamma (\mathcal{C}, \mathcal{I}^\bullet ) \to \Gamma (\mathcal{C}, \mathcal{J}^\bullet )$ (2) is given by $\Gamma (U, \mathcal{I}^\bullet ) \to \Gamma (U, \mathcal{J}^\bullet )$ and the map in (3) is given by $f_*\mathcal{I}^\bullet \to f_*\mathcal{J}^\bullet $. To show that these maps are isomorphisms, it suffices to prove they induce isomorphisms on cohomology groups and cohomology sheaves. By Lemmas 21.20.2 and 21.20.6 it suffices to show that the map

\[ H^0(\mathcal{C}, K) \longrightarrow H^0(\mathcal{C}, K_{ab}) \]

is an isomorphism. Observe that

\[ H^0(\mathcal{C}, K) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {C})}(\mathcal{O}_\mathcal {C}, K) \]

and similarly for the other group. Choose any complex $\mathcal{K}^\bullet $ of $\mathcal{O}_\mathcal {C}$-modules representing $K$. By construction of the derived category as a localization we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {C})}(\mathcal{O}_\mathcal {C}, K) = \mathop{\mathrm{colim}}\nolimits _{s : \mathcal{F}^\bullet \to \mathcal{O}_\mathcal {C}} \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_\mathcal {C})}(\mathcal{F}^\bullet , \mathcal{K}^\bullet ) \]

where the colimit is over quasi-isomorphisms $s$ of complexes of $\mathcal{O}_\mathcal {C}$-modules. Similarly, we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\underline{\mathbf{Z}}_\mathcal {C})} (\underline{\mathbf{Z}}_\mathcal {C}, K) = \mathop{\mathrm{colim}}\nolimits _{s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal {C}} \mathop{\mathrm{Hom}}\nolimits _{K(\underline{\mathbf{Z}}_\mathcal {C})} (\mathcal{G}^\bullet , \mathcal{K}^\bullet ) \]

Next, we observe that the quasi-isomorphisms $s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal {C}$ with $\mathcal{G}^\bullet $ bounded above complex of flat $\underline{\mathbf{Z}}_\mathcal {C}$-modules is cofinal in the system. (This follows from Modules on Sites, Lemma 18.28.8 and Derived Categories, Lemma 13.15.4; see discussion in Section 21.17.) Hence we can construct an inverse to the map $H^0(\mathcal{C}, K) \longrightarrow H^0(\mathcal{C}, K_{ab})$ by representing an element $\xi \in H^0(\mathcal{C}, K_{ab})$ by a pair

\[ (s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal {C}, a : \mathcal{G}^\bullet \to \mathcal{K}^\bullet ) \]

with $\mathcal{G}^\bullet $ a bounded above complex of flat $\underline{\mathbf{Z}}_\mathcal {C}$-modules and sending this to

\[ (\mathcal{G}^\bullet \otimes _{\underline{\mathbf{Z}}_\mathcal {C}} \mathcal{O}_\mathcal {C} \to \mathcal{O}_\mathcal {C}, \mathcal{G}^\bullet \otimes _{\underline{\mathbf{Z}}_\mathcal {C}} \mathcal{O}_\mathcal {C} \to \mathcal{K}^\bullet ) \]

The only thing to note here is that the first arrow is a quasi-isomorphism by Lemmas 21.17.12 and 21.17.8. We omit the detailed verification that this construction is indeed an inverse.
$\square$

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