We observe that property (3) is a consequence of properties (1) and (2). Namely, if $s$ is a nonzero local section of $F(G(\mathcal{N}))$ with $\text{d}(s) = 0$, then $s$ cannot be in the image of $\mathcal{N} \to F(G(\mathcal{N}))$. Hence we can write the image $\overline{s}$ of $s$ in the cokernel as $\overline{\text{d}}(s')$ for some local section $s'$ of $\mathcal{N}$. Then we see that $s = \text{d}(s')$ because the difference $s - \text{d}(s')$ is still in the kernel of $\text{d}$ and is contained in the image of the counit. \medskip\noindent Let us write temporarily $\mathcal{A}_{gr}$, respectively $\mathcal{A}_{dg}$ the sheaf $\mathcal{A}$ viewed as a (right) graded module over itself, respectively as a (right) differential graded module over itself. The most important case of the lemma is to understand what is $G(\mathcal{A}_{gr})$. Of course $G(\mathcal{A}_{gr})$ is the object of $\text{Mod}_{(\mathcal{A}, \text{d})}$ representing the functor $$ \mathcal{M} \longmapsto \Hom_{\text{Mod}_\mathcal{A}}(\mathcal{A}_{gr}, F(\mathcal{M})) = \Gamma(\mathcal{C}, \mathcal{M}) $$ By Remark \ref{remark-cone-identity} we see that this functor represented by $C[-1]$ where $C$ is the cone on the identity of $\mathcal{A}_{dg}$. We have a short exact sequence $$ 0 \to \mathcal{A}_{dg}[-1] \to C[-1] \to \mathcal{A}_{dg} \to 0 $$ in $\text{Mod}_{(\mathcal{A}, \text{d})}$ which is split by the counit $\mathcal{A}_{gr} \to F(C[-1])$ in $\text{Mod}_\mathcal{A}$. Thus $G(\mathcal{A}_{gr})$ satisfies properties (1) and (2). \medskip\noindent Let $U$ be an object of $\mathcal{C}$. Denote $j_U : \mathcal{C}/U \to \mathcal{C}$ the localization morphism. Denote $\mathcal{A}_U$ the restriction of $\mathcal{A}$ to $U$. We will use the notation $\mathcal{A}_{U, gr}$ to denote $\mathcal{A}_U$ viewed as a graded $\mathcal{A}_U$-module. Denote $F_U : \text{Mod}_{(\mathcal{A}_U, \text{d})} \to \text{Mod}_{\mathcal{A}_U}$ the forgetful functor and denote $G_U$ its adjoint. Then we have the commutative diagrams $$ \vcenter{ \xymatrix{ \text{Mod}_{(\mathcal{A}, \text{d})} \ar[d]_{j_U^*} \ar[r]_F & \text{Mod}_\mathcal{A} \ar[d]^{j_U^*} \\ \text{Mod}_{(\mathcal{A}_U, \text{d})} \ar[r]^{F_U} & \text{Mod}_{\mathcal{A}_U} } } \quad\text{and}\quad \vcenter{ \xymatrix{ \text{Mod}_{(\mathcal{A}_U, \text{d})} \ar[r]_{F_U} \ar[d]_{j_{U!}} & \text{Mod}_{\mathcal{A}_U} \ar[d]^{j_{U!}} \\ \text{Mod}_{(\mathcal{A}, \text{d})} \ar[r]^F & \text{Mod}_\mathcal{A} } } $$ by the construction of $j^*_U$ and $j_{U!}$ in Sections \ref{section-functoriality-graded}, \ref{section-functoriality-dg}, \ref{section-localize-graded}, and \ref{section-localize-dg}. By uniqueness of adjoints we obtain $j_{U!} \circ G_U = G \circ j_{U!}$. Since $j_{U!}$ is an exact functor, we see that the properties (1) and (2) for the counit $\mathcal{A}_{U, gr} \to F_U(G_U(\mathcal{A}_{U, gr}))$ which we've seen in the previous part of the proof imply properties (1) and (2) for the counit $j_{U!}\mathcal{A}_{U, gr} \to F(G(j_{U!}\mathcal{A}_{U, gr})) = j_{U!}F_U(G_U(\mathcal{A}_{U, gr}))$. \medskip\noindent In the proof of Lemma \ref{lemma-gm-grothendieck-abelian} we have seen that any object of $\text{Mod}_\mathcal{A}$ is a quotient of a direct sum of copies of $j_{U!}\mathcal{A}_{U, gr}$. Since $G$ is a left adjoint, we see that $G$ commutes with direct sums. Thus properties (1) and (2) hold for direct sums of objects for which they hold. Thus we see that every object $\mathcal{N}$ of $\text{Mod}_\mathcal{A}$ fits into an exact sequence $$ \mathcal{N}_1 \to \mathcal{N}_0 \to \mathcal{N} \to 0 $$ such that (1) and (2) hold for $\mathcal{N}_1$ and $\mathcal{N}_0$. We leave it to the reader to deduce (1) and (2) for $\mathcal{N}$ using that $G$ is right exact.