Pullback-direct-image adjunction unit (${\eta }_{\mathcal{H}}:\mathcal{H}\to f_{\ast }{f}^{\ast }\mathcal{H}$)

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 ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$.