In this proof $\alpha, \beta, \gamma, \ldots$ represent elements of $E$. Choose a K-injective complex $I_\alpha^\bullet$ on $W_\alpha$ representing $K_\alpha$. For $\beta < \alpha$ denote $j_{\beta, \alpha} : W_\beta \to W_\alpha$ the inclusion morphism. By transfinite induction, we will construct for all $\beta < \alpha$ a map of complexes $$ \tau_{\beta, \alpha} : (j_{\beta, \alpha})_!I_\beta^\bullet \longrightarrow I_\alpha^\bullet $$ representing the adjoint to the inverse of the isomorphism $\rho^\alpha_\beta : K_\alpha|_{W_\beta} \to K_\beta$. Moreover, we will do this in such that for $\gamma < \beta < \alpha$ we have $$ \tau_{\gamma, \alpha} = \tau_{\beta, \alpha} \circ (j_{\beta, \alpha})_!\tau_{\gamma, \beta} $$ as maps of complexes. Namely, suppose already given $\tau_{\gamma, \beta}$ composing correctly for all $\gamma < \beta < \alpha$. If $\alpha = \alpha' + 1$ is a successor, then we choose any map of complexes $$ (j_{\alpha', \alpha})_!I_{\alpha'}^\bullet \to I_\alpha^\bullet $$ which is adjoint to the inverse of the isomorphism $\rho^\alpha_{\alpha'} : K_\alpha|_{W_{\alpha'}} \to K_{\alpha'}$ (possible because $I_\alpha^\bullet$ is K-injective) and for any $\beta < \alpha'$ we set $$ \tau_{\beta, \alpha} = \tau_{\alpha', \alpha} \circ (j_{\alpha', \alpha})_!\tau_{\beta, \alpha'} $$ If $\alpha$ is not a successor, then we can consider the complex on $W_\alpha$ given by $$ C^\bullet = \colim_{\beta < \alpha} (j_{\beta, \alpha})_!I_\beta^\bullet $$ (termwise colimit) where the transition maps of the sequence are given by the maps $\tau_{\beta', \beta}$ for $\beta' < \beta < \alpha$. We claim that $C^\bullet$ represents $K_\alpha$. Namely, for $\beta < \alpha$ the restriction of the coprojection $(j_{\beta, \alpha})_!I_\beta^\bullet \to C^\bullet$ gives a map $$ \sigma_\beta : I_\beta^\bullet \longrightarrow C^\bullet|_{W_\beta} $$ which is a quasi-isomorphism: if $x \in W_\beta$ then looking at stalks we get $$ (C^\bullet)_x = \colim_{\beta' < \alpha} \left((j_{\beta', \alpha})_!I_{\beta'}^\bullet\right)_x = \colim_{\beta \leq \beta' < \alpha} (I_{\beta'}^\bullet)_x \longleftarrow (I_\beta^\bullet)_x $$ which is a quasi-isomorphism. Here we used that taking stalks commutes with colimits, that filtered colimits are exact, and that the maps $(I_\beta^\bullet)_x \to (I_{\beta'}^\bullet)_x$ are quasi-isomorphisms for $\beta \leq \beta' < \alpha$. Hence $(C^\bullet, \sigma_\beta^{-1})$ is a solution to the system $(\{K_\beta\}_{\beta < \alpha}, \{\rho^\beta_{\beta'}\}_{\beta' < \beta < \alpha})$. Since $(K_\alpha, \rho^\alpha_\beta)$ is another solution we obtain a unique isomorphism $\sigma : K_\alpha \to C^\bullet$ in $D(\mathcal{O}_{W_\alpha})$ compatible with all our maps, see Lemma \ref{lemma-solution-in-finite-case} (this is where we use the vanishing of negative ext groups). Choose a morphism $\tau : C^\bullet \to I_\alpha^\bullet$ of complexes representing $\sigma$. Then we set $$ \tau_{\beta, \alpha} = \tau|_{W_\beta} \circ \sigma_\beta $$ to get the desired maps. Finally, we take $K$ to be the object of the derived category represented by the complex $$ K^\bullet = \colim_{\alpha \in E} (W_\alpha \to X)_!I_\alpha^\bullet $$ where the transition maps are given by our carefully constructed maps $\tau_{\beta, \alpha}$ for $\beta < \alpha$. Arguing exactly as above we see that for all $\alpha$ the restriction of the coprojection determines an isomorphism $$ K|_{W_\alpha} \longrightarrow K_\alpha $$ compatible with the given maps $\rho^\alpha_\beta$.