The case of abelian sheaves follows immediately from the case of sheaves of sets as the functor $a^{-1}$ commutes with products. In the rest of the proof we work with sheaves of sets. Observe that $a^{-1}\mathcal{F}$ is cartesian for $\mathcal{F}$ in $\Sh(\mathcal{C})$ by Lemma \ref{lemma-augmentation-cartesian-module}. It suffices to show that the adjunction map $\mathcal{F} \to a_*a^{-1}\mathcal{F}$ is an isomorphism $\mathcal{F}$ in $\Sh(\mathcal{C})$ and that for a cartesian sheaf $\mathcal{G}$ on $(\mathcal{C}/K)_{total}$ the adjunction map $a^{-1}a_*\mathcal{G} \to \mathcal{G}$ is an isomorphism. \medskip\noindent Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Recall that $a_*a^{-1}\mathcal{F}$ is the equalizer of the two maps $a_{0, *}a_0^{-1}\mathcal{F} \to a_{1, *}a_1^{-1}\mathcal{F}$, see Lemma \ref{lemma-comparison}. By Lemma \ref{lemma-push-pull-localization} $$ a_{0, *}a_0^{-1}\mathcal{F} = \SheafHom(F(K_0)^\#, \mathcal{F}) \quad\text{and}\quad a_{1, *}a_1^{-1}\mathcal{F} = \SheafHom(F(K_1)^\#, \mathcal{F}) $$ On the other hand, we know that $$ \xymatrix{ F(K_1)^\# \ar@<1ex>[r] \ar@<-1ex>[r] & F(K_0)^\# \ar[r] & \text{final object }*\text{ of }\Sh(\mathcal{C}) } $$ is a coequalizer diagram in sheaves of sets by definition of a hypercovering. Thus it suffices to prove that $\SheafHom(-, \mathcal{F})$ transforms coequalizers into equalizers which is immediate from the construction in Sites, Section \ref{sites-section-glueing-sheaves}. \medskip\noindent Let $\mathcal{G}$ be a cartesian sheaf on $(\mathcal{C}/K)_{total}$. We will show that $\mathcal{G} = a^{-1}\mathcal{F}$ for some sheaf $\mathcal{F}$ on $\mathcal{C}$. This will finish the proof because then $a^{-1}a_*\mathcal{G} = a^{-1}a_*a^{-1}\mathcal{F} = a^{-1}\mathcal{F} = \mathcal{G}$ by the result of the previous paragraph. Set $\mathcal{K}_n = F(K_n)^\#$ for $n \geq 0$. Then we have maps of sheaves $$ \xymatrix{ \mathcal{K}_2 \ar@<1ex>[r] \ar@<0ex>[r] \ar@<-1ex>[r] & \mathcal{K}_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{K}_0 } $$ coming from the fact that $K$ is a simplicial semi-representable object. The fact that $K$ is a hypercovering means that $$ \mathcal{K}_1 \to \mathcal{K}_0 \times \mathcal{K}_0 \quad\text{and}\quad \mathcal{K}_2 \to \left(\text{cosq}_1( \xymatrix{ \mathcal{K}_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{K}_0 \ar[l] })\right)_2 $$ are surjective maps of sheaves. Using the description of cartesian sheaves on $(\mathcal{C}/K)_{total}$ given in Lemma \ref{lemma-characterize-cartesian} and using the description of $\Sh(\mathcal{C}/K_n)$ in Lemma \ref{lemma-localize-compare} we find that our problem can be entirely formulated\footnote{Even though it does not matter what the precise formulation is, we spell it out: the problem is to show that given an object $\mathcal{G}_0/\mathcal{K}_0$ of $\Sh(\mathcal{C})/\mathcal{K}_0$ and an isomorphism $$ \alpha : \mathcal{G}_0 \times_{\mathcal{K}_0, \mathcal{K}(\delta^1_1)} \mathcal{K}_1 \to \mathcal{G}_0 \times_{\mathcal{K}_0, \mathcal{K}(\delta^1_0)} \mathcal{K}_1 $$ over $\mathcal{K}_1$ satisfying a cocycle condtion in $\Sh(\mathcal{C})/\mathcal{K}_2$, there exists $\mathcal{F}$ in $\Sh(\mathcal{C})$ and an isomorphism $\mathcal{F} \times \mathcal{K}_0 \to \mathcal{G}_0$ over $\mathcal{K}_0$ compatible with $\alpha$.} in terms of \begin{enumerate} \item the topos $\Sh(\mathcal{C})$, and \item the simplicial object $\mathcal{K}$ in $\Sh(\mathcal{C})$ whose terms are $\mathcal{K}_n$. \end{enumerate} Thus, after replacing $\mathcal{C}$ by a different site $\mathcal{C}'$ as in Sites, Lemma \ref{sites-lemma-topos-good-site}, we may assume $\mathcal{C}$ has all finite limits, the topology on $\mathcal{C}$ is subcanonical, a family $\{V_j \to V\}$ of morphisms of $\mathcal{C}$ is a covering if and only if $\coprod h_{V_j} \to V$ is surjective, and there exists a simplicial object $U$ of $\mathcal{C}$ such that $\mathcal{K}_n = h_{U_n}$ as simplicial sheaves. Working backwards through the equivalences we may assume $K_n = \{U_n\}$ for all $n$. \medskip\noindent Let $X$ be the final object of $\mathcal{C}$. Then $\{U_0 \to X\}$ is a covering, $\{U_1 \to U_0 \times U_0\}$ is a covering, and $\{U_2 \to (\text{cosq}_1 \text{sk}_1 U)_2\}$ is a covering. Let us use $d^n_i : U_n \to U_{n - 1}$ and $s^n_j : U_n \to U_{n + 1}$ the morphisms corresponding to $\delta^n_i$ and $\sigma^n_j$ as in Simplicial, Definition \ref{simplicial-definition-face-degeneracy}. By abuse of notation, given a morphism $c : V \to W$ of $\mathcal{C}$ we denote the morphism of topoi $c : \Sh(\mathcal{C}/V) \to \Sh(\mathcal{C}/W)$ by the same letter. Now $\mathcal{G}$ is given by a sheaf $\mathcal{G}_0$ on $\mathcal{C}/U_0$ and an isomorphism $\alpha : (d^1_1)^{-1}\mathcal{G}_0 \to (d^1_0)^{-1}\mathcal{G}_0$ satisfying the cocycle condition on $\mathcal{C}/U_2$ formulated in Lemma \ref{lemma-characterize-cartesian}. Since $\{U_2 \to (\text{cosq}_1 \text{sk}_1 U)_2\}$ is a covering, the corresponding pullback functor on sheaves is faithful (small detail omitted). Hence we may replace $U$ by $\text{cosk}_1 \text{sk}_1 U$, because this replaces $U_2$ by $(\text{cosq}_1 \text{sk}_1 U)_2$ and leaves $U_1$ and $U_0$ unchanged. Then $$ (d^2_0, d^2_1, d^2_2) : U_2 \to U_1 \times U_1 \times U_1 $$ is a monomorphism whose its image on $T$-valued points is described in Simplicial, Lemma \ref{simplicial-lemma-work-out}. In particular, there is a morphism $c$ fitting into a commutative diagram $$ \xymatrix{ U_1 \times_{(d^1_1, d^1_0), U_0 \times U_0, (d^1_1, d^1_0)} U_1 \ar[d] \ar[rr]_c & & U_2 \ar[d] \\ U_1 \times U_1 \ar[rr]^{(\text{pr}_1, \text{pr}_2, s^0_0 \circ d^1_1 \circ \text{pr}_1)} & & U_1 \times U_1 \times U_1 } $$ as going around the other way defines a point of $U_2$. Pulling back the cocycle condition for $\alpha$ on $U_2$ translates into the condition that the pullbacks of $\alpha$ via the projections to $U_1 \times_{(d^1_1, d^1_0), U_0 \times U_0, (d^1_1, d^1_0)} U_1$ are the same as the pullback of $\alpha$ via $s^0_0 \circ d^1_1 \circ \text{pr}_1$ is the identity map (namely, the pullback of $\alpha$ by $s^0_0$ is the identity). By Sites, Lemma \ref{sites-lemma-glue-maps} this means that $\alpha$ comes from an isomorphism $$ \alpha' : \text{pr}_1^{-1}\mathcal{G}_0 \to \text{pr}_2^{-1}\mathcal{G}_0 $$ of sheaves on $\mathcal{C}/U_0 \times U_0$. Then finally, the morphism $U_2 \to U_0 \times U_0 \times U_0$ is surjective on associated sheaves as is easily seen using the surjectivity of $U_1 \to U_0 \times U_0$ and the description of $U_2$ given above. Therefore $\alpha'$ satisfies the cocycle condition on $U_0 \times U_0 \times U_0$. The proof is finished by an application of Sites, Lemma \ref{sites-lemma-mapping-property-glue} to the covering $\{U_0 \to X\}$.