Is there a nice explicit description for the unit and counit of the inverse image/direct image adjunction $f^{-1} \dashv f_*$ between sheaves of rings (and in the version $f^* \dashv f_*$ for $\mathcal{O}_X$-modules)? It is said these come from natural maps, but I can only see they exist because I can argue we should have a hom-set adjunction, and that iso is constructed by using the universal property of the colimit of which $f^{-1}$ is a sheafification (in the sheaf case).
Here I'm using the definition that given a morphism $f:(X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$ of ringed spaces, and sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}$ on $Y$ (resp. of $\mathcal{O}_X$ and $\mathcal{O}_Y$ modules), then $f^{-1}\mathcal{G}$ is the shefification of the presheaf $U \mapsto \ colim_{f(U) \subseteq V}\ \mathcal{G}(V)$ and in the case of $\mathcal{O}_X$-modules we define $f^*\mathcal{G} = f^{-1}\mathcal{G} \otimes_{f^{-1} \mathcal{O}_Y} \mathcal{O}_X$.