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In lots of different cases (Verdier duality, Grothendieck duality, étale cohomology, ...) the very existence of a (right) adjoint to the sheaf functor $f_!$ gives useful information. (I'm going to ignore whether the functors are derived or not.)

As it appears, in many of those same cases, the functor $f^*$ also has a left adjoint, which seems to be denoted as $f_\sharp$ in some places. (Examples: this paper talks about this in Grothendieck duality and these notes talk about this for $f$ smooth.)

Beyond implying that $f^*$ is exact, which we already often know, does this functor gives any meaningful information? I would appreciate knowing anything about it!

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    $\begingroup$ In the derived world (that is for exact functors between presentable stable ∞-categories) the existence of a left adjoint to $f^*$ is equivalent to saying that $f^*$ commutes with arbitrary products. In general, the existence of $f_#$ is related to $f^*$ commuting with limits (for example, in motivic homotopy theory, when $f_#$ exists, then $f^*$ commutes with loopspaces, which is often very useful). $\endgroup$
    – Denis Nardin
    Commented Jan 11, 2022 at 15:23

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