Pullback-direct-image adjunction unit
= Pullback-direct-image adjunction unit
{title2=$\eta_{\mathcal H}:\mathcal H\to f_*f^*\mathcal H$}
For a <morphism of ringed spaces>, the <pullback of a sheaf of modules> is left adjoint to the <direct image sheaf> functor. The unit sends a local section $h$ to $1\otimes h$ after inverse image. On the <structure sheaf> it becomes the given <ring homomorphism> $\mathcal O_Y\to f_*\mathcal O_X$.