We are going to use Derived Categories, Proposition \ref{derived-proposition-left-derived-exists} to construct $Lg_!$. To do this we have to verify assumptions (1), (2), (3), (4), and (5) of that proposition. First, since $g_!$ is a left adjoint we see that it is right exact and commutes with all colimits, so (5) holds. Conditions (3) and (4) hold because the category of modules on a ringed site is a Grothendieck abelian category. Let $\mathcal{P} \subset \Ob(\textit{Mod}(\mathcal{O}_\mathcal{C}))$ be the collection of $\mathcal{O}_\mathcal{C}$-modules which are direct sums of modules of the form $j_{U!}\mathcal{O}_U$. Note that $g_!j_{U!}\mathcal{O}_U = j_{u(U)!} \mathcal{O}_{u(U)}$, see proof of Modules on Sites, Lemma \ref{sites-modules-lemma-lower-shriek-modules}. Every $\mathcal{O}_\mathcal{C}$-module is a quotient of an object of $\mathcal{P}$, see Modules on Sites, Lemma \ref{sites-modules-lemma-module-quotient-flat}. Thus (1) holds. Finally, we have to prove (2). Let $\mathcal{K}^\bullet$ be a bounded above acyclic complex of $\mathcal{O}_\mathcal{C}$-modules with $\mathcal{K}^n \in \mathcal{P}$ for all $n$. We have to show that $g_!\mathcal{K}^\bullet$ is exact. To do this it suffices to show, for every injective $\mathcal{O}_\mathcal{D}$-module $\mathcal{I}$ that $$ \Hom_{D(\mathcal{O}_\mathcal{D})}( g_!\mathcal{K}^\bullet, \mathcal{I}[n]) = 0 $$ for all $n \in \mathbf{Z}$. Since $\mathcal{I}$ is injective we have \begin{align*} \Hom_{D(\mathcal{O}_\mathcal{D})}( g_!\mathcal{K}^\bullet, \mathcal{I}[n]) & = \Hom_{K(\mathcal{O}_\mathcal{D})}( g_!\mathcal{K}^\bullet, \mathcal{I}[n]) \\ & = H^n(\Hom_{\mathcal{O}_\mathcal{D}}( g_!\mathcal{K}^\bullet, \mathcal{I})) \\ & = H^n(\Hom_{\mathcal{O}_\mathcal{C}}( \mathcal{K}^\bullet, g^{-1}\mathcal{I})) \end{align*} the last equality by the adjointness of $g_!$ and $g^{-1}$. \medskip\noindent The vanishing of this group would be clear if $g^{-1}\mathcal{I}$ were an injective $\mathcal{O}_\mathcal{C}$-module. But $g^{-1}\mathcal{I}$ isn't necessarily an injective $\mathcal{O}_\mathcal{C}$-module as $g_!$ isn't exact in general. We do know that $$ \Ext^p_{\mathcal{O}_\mathcal{C}}( j_{U!}\mathcal{O}_U, g^{-1}\mathcal{I}) = H^p(U, g^{-1}\mathcal{I}) = 0 \text{ for }p \geq 1 $$ Here the first equality follows from $\Hom_{\mathcal{O}_\mathcal{C}}(j_{U!}\mathcal{O}_U, \mathcal{H}) = \mathcal{H}(U)$ and taking derived functors and the vanishing of $H^p(U, g^{-1}\mathcal{I})$ for $p > 0$ and $U \in \Ob(\mathcal{C})$ follows from Lemma \ref{lemma-pullback-injective-pre-limp}. Since each $\mathcal{K}^{-q}$ is a direct sum of modules of the form $j_{U!}\mathcal{O}_U$ we see that $$ \Ext^p_{\mathcal{O}_\mathcal{C}}(\mathcal{K}^{-q}, g^{-1}\mathcal{I}) = 0 \text{ for }p \geq 1\text{ and all }q $$ Let us use the spectral sequence (see Example \ref{example-hom-complex-into-sheaf}) $$ E_1^{p, q} = \Ext^p_{\mathcal{O}_\mathcal{C}}( \mathcal{K}^{-q}, g^{-1}\mathcal{I}) \Rightarrow \Ext^{p + q}_{\mathcal{O}_\mathcal{C}}( \mathcal{K}^\bullet, g^{-1}\mathcal{I}) = 0. $$ Note that the spectral sequence abuts to zero as $\mathcal{K}^\bullet$ is acyclic (hence vanishes in the derived category, hence produces vanishing ext groups). By the vanishing of higher exts proved above the only nonzero terms on the $E_1$ page are the terms $E_1^{0, q} = \Hom_{\mathcal{O}_\mathcal{C}}( \mathcal{K}^{-q}, g^{-1}\mathcal{I})$. We conclude that the complex $\Hom_{\mathcal{O}_\mathcal{C}}( \mathcal{K}^\bullet, g^{-1}\mathcal{I})$ is acyclic as desired. \medskip\noindent Thus the left derived functor $Lg_!$ exists. It is left adjoint to $g^{-1} = g^* = Rg^* = Lg^*$, i.e., we have \begin{equation} \label{equation-to-prove} \Hom_{D(\mathcal{O}_\mathcal{C})}(K, g^*L) = \Hom_{D(\mathcal{O}_\mathcal{D})}(Lg_!K, L) \end{equation} by Derived Categories, Lemma \ref{derived-lemma-derived-adjoint-functors}. This finishes the proof.