Inverse image is left adjoint to direct image on sheaves

Let $\alpha :f^{-1}\mathcal{G}\to \mathcal{F}$ be a sheaf morphism. By [F2] and [L1], $\alpha$ corresponds uniquely to a presheaf morphism $\tilde{\alpha }:f_{\mathrm{p}}\mathcal{G}\to \mathcal{F}$.

Conversely, let $\beta :\mathcal{G}\to f_{\ast }\mathcal{F}$ be a sheaf morphism. For an open $U\subseteq X$, represent an element of $(f_{\mathrm{p}}\mathcal{G})(U)$ by a pair $(V,s)$ with $f(U)\subseteq V$ and $s\in \mathcal{G}(V)$, and define ${\tilde{\beta }}_U([V,s]):={\beta }_V(s){\mathrm{\mid }}_U\in \mathcal{F}(U),$ where the restriction is taken from $\mathcal{F}(f^{-1}(V))$ to $\mathcal{F}(U)$. If $(V,s)$ and $(V^{\mathrm{\prime }},s^{\mathrm{\prime }})$ represent the same colimit class, then on some smaller open neighbourhood $W\subseteq V\cap V^{\mathrm{\prime }}$ of $f(U)$ one has $s{\mathrm{\mid }}_W=s^{\mathrm{\prime }}{\mathrm{\mid }}_W$, so ${\beta }_V(s){\mathrm{\mid }}_U={\beta }_{V^{\mathrm{\prime }}}(s^{\mathrm{\prime }}){\mathrm{\mid }}_U$. Thus $\tilde{\beta }$ is well defined and natural in $U$.

For an open set $V\subseteq Y$ and a section $s\in \mathcal{G}(V)$, let $[V,s]$ denote its class in $(f_{\mathrm{p}}\mathcal{G})(f^{-1}(V))$, and define ${\beta }_V(s):={\tilde{\alpha }}_{f^{-1}(V)}([V,s])\in \mathcal{F}(f^{-1}(V))=(f_{\ast }\mathcal{F})(V)\mathrm{.}$ If $V^{\mathrm{\prime }}\subseteq V$, naturality of $\tilde{\alpha }$ shows ${\beta }_{V^{\mathrm{\prime }}}(s{\mathrm{\mid }}_{V^{\mathrm{\prime }}})={\beta }_V(s){\mathrm{\mid }}_{V^{\mathrm{\prime }}}$, so the ${\beta }_V$ define a morphism $\beta :\mathcal{G}\to f_{\ast }\mathcal{F}$.

By [F2] and [L1], the presheaf morphism $\tilde{\beta }:f_{\mathrm{p}}\mathcal{G}\to \mathcal{F}$ factors uniquely through a sheaf morphism $\alpha :f^{-1}\mathcal{G}\to \mathcal{F}$.

The constructions of steps 2.1 and 2.2 are inverse, because both are recovered from the same formula on representative classes $[V,s]$. Therefore ${\mathrm{Hom}}_X(f^{-1}\mathcal{G},\mathcal{F})\cong {\mathrm{Hom}}_Y(\mathcal{G},f_{\ast }\mathcal{F})$ naturally in both sheaves.