Let $X$ be a topological space, $i:U\to X$, I want to show the adjunction $i_{!}\dashv i^{-1}$. To do so I aim to find a natural isomorphism $\operatorname{Hom}(i_{!}\mathcal F,\mathcal G)\cong \operatorname{Hom}(\mathcal F,i^{-1}G)$ for sheaves of abelian groups $\mathcal F,\mathcal G$ on $U,X$ respectively.
If $\varphi\colon i_{!}\mathcal F\to \mathcal G$ is a morphism with for $W\subseteq X$ $$\varphi(W)\colon i_{!}\mathcal F(W)\to \mathcal G(W),\quad (s\colon W\to \bigcup_{p\in W}(i_{!}F)_p)\mapsto \varphi(W)(s)$$
we define a morphism $\psi\colon \mathcal F\to i^{-1}\mathcal G$ defined for $V\subseteq U$ by $$\varphi(V)\colon \mathcal F(V)\to (i^{-1}\mathcal G)(V),\quad t\mapsto s_t$$ with $$s_t\colon V\to \bigcup_{p\in V}(i^{-1}\mathcal G)_p=\bigcup_{p\in V}\mathcal G_p,\quad x\mapsto (\varphi(V)(t_x))_x$$
Then sending $\varphi$ to $\psi$ gives a morphism from $\operatorname{Hom}(i_{!}\mathcal F,\mathcal G)$ to $\operatorname{Hom}(\mathcal F,i^{-1}G)$. Is this a good candidate ?
Now for the other direction, let $\psi\colon \mathcal F\to i^{-1}\mathcal G $ be defined for $V\subseteq U$ by $$\psi(V)\colon \mathcal F(V)\to i^{-1}\mathcal G(V),\quad t\mapsto (\psi_t\colon V\to \bigcup_{p\in V}G_p)$$ To get a morphism $i_{!}\mathcal F\to \mathcal G$ I want to define a morphism $i_{!}\mathcal F^{pre}\to \mathcal G$ and use the universal property of the associated sheaf so define $\varphi\colon i_{!}\mathcal F^{pre}\to \mathcal G$ by $\varphi(W)=0$ if $W\nsubseteq U$ (in this case $i_{!}\mathcal F^{pre}(W)=0$ so we have no choice) and if $W\subseteq U$ $$\varphi(W)\colon \mathcal F(W)=i_{!}\mathcal F^{pre}(W)\to \mathcal G(W)$$ by... I don't know. I'm not sure on how to use $\psi$ since it takes a section to a map $\psi_t\colon W\to \bigcup_{p\in W}G_p$, but then what section of $\mathcal G(W)$ can we get ? Do you know any reference showing this ?