Note that by Lemma \ref{lemma-adjointness-pullback-pushforward-stacks} we have an equivalence of categories $$ \Mor_{\textit{Stacks}/\mathcal{D}}(f^{-1}\mathcal{S}, f^{-1}\mathcal{S}') = \Mor_{\textit{Stacks}/\mathcal{C}}(\mathcal{S}, f_*f^{-1}\mathcal{S}') $$ Hence (2) follows from (1). \medskip\noindent To prove (1) we are going to use Lemma \ref{lemma-characterize-essentially-surjective-when-ff}. This lemma tells us that we have to show that $can : \mathcal{S} \to f_*f^{-1}\mathcal{S}$ is fully faithful and that all objects of $f_*f^{-1}\mathcal{S}$ are locally in the essential image. \medskip\noindent We quickly describe the functor $can$, see proof of Lemma \ref{lemma-adjointness-pullback-pushforward}. To do this we introduce the functor $c'' : \mathcal{S} \to u_{pp}\mathcal{S}$ defined by $c''(x/U) = (U, \text{id} : u(U) \to u(U), x)$, and $c''(\alpha/a) = (a, u(a), \alpha)$. We set $c' : \mathcal{S} \to u_p\mathcal{S}$ equal to the composition of $c''$ and the canonical functor $u_{pp}\mathcal{S} \to u_p\mathcal{S}$. We set $c : \mathcal{S} \to f^{-1}\mathcal{S}$ equal to the composition of $c'$ and the canonical functor $u_p\mathcal{S} \to f^{-1}\mathcal{S}$. Then $can : \mathcal{S} \to f_*f^{-1}\mathcal{S}$ is the functor which to $x/U$ associates the pair $(U, c(x))$ and to $\alpha/a$ the morphism $(a, c(\alpha))$. \medskip\noindent Fully faithfulness. To prove this we are going to use Lemma \ref{lemma-characterize-ff}. Let $U \in \Ob(\mathcal{C})$. Let $x, y \in \mathcal{S}_U$. First off, as $u$ is fully faithful, we have $$ \Mor_{(f_*f^{-1}\mathcal{S})_U}(can(x), can(y)) = \Mor_{(f^{-1}\mathcal{S})_{u(U)}}(c(x), c(y)) $$ directly from the definition of $f_*$. Similar holds after pulling back to any $U'/U$. Because $f^{-1}\mathcal{S}$ is the stackification of $u_p\mathcal{S}$, and since $u$ is continuous and cocontinuous the presheaf $$ U'/U \longmapsto \Mor_{(f^{-1}\mathcal{S})_{u(U')}}(c(x|_{U'}), c(y|_{U'})) $$ is the sheafification of the presheaf $$ U'/U \longmapsto \Mor_{(u_p\mathcal{S})_{u(U')}}(c'(x|_{U'}), c'(y|_{U'})) $$ Hence to finish the proof of fully faithfulness it suffices to show that for any $U$ and $x, y$ the map $$ \Mor_{\mathcal{S}_U}(x, y) \longrightarrow \Mor_{(u_p\mathcal{S})_U}(c'(x), c'(y)) $$ is bijective. A morphism $f : x \to y$ in $u_p\mathcal{S}$ over $u(U)$ is given by an equivalence class of diagrams $$ \xymatrix{ (U', \phi : u(U) \to u(U'), x') \ar[d]_{(c, \text{id}_{u(U)}, \gamma)} \ar[r]_{(a, b, \alpha)} & (U, \text{id} : u(U) \to u(U), y) \\ (U, \text{id} : u(U) \to u(U), x) } $$ with $\gamma$ strongly cartesian and $b = \text{id}_{u(U)}$. But since $u$ is fully faithful we can write $\phi = u(c')$ for some morphism $c' : U \to U'$ and then we see that $a \circ c' = \text{id}_U$ and $c \circ c' = \text{id}_{U'}$. Because $\gamma$ is strongly cartesian we can find a morphism $\gamma' : x \to x'$ lifting $c'$ such that $\gamma \circ \gamma' = \text{id}_x$. By definition of the equivalence classes defining morphisms in $u_p\mathcal{S}$ it follows that the morphism $$ \xymatrix{ (U, \text{id} : u(U) \to u(U), x) \ar[rr]_{(\text{id}, \text{id}, \alpha \circ \gamma')} & & (U, \text{id} : u(U) \to u(U), y) } $$ of $u_{pp}\mathcal{S}$ induces the morphism $f$ in $u_p\mathcal{S}$. This proves that the map is surjective. We omit the proof that it is injective. \medskip\noindent Finally, we have to show that any object of $f_*f^{-1}\mathcal{S}$ locally comes from an object of $\mathcal{S}$. This is clear from the constructions (details omitted).