TITLE: How do the direct and inverse image sheaf functors interact with homotopy?
QUESTION [10 upvotes]: This is a crosspost of this MSE question.
The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$) form a well-known adjunction for $\mathsf{Set}$-valued sheaves (I think also for sheaves valued in algebraic categories).

Is there anything interesting to be said about $f^\ast,g^\ast$ and
  $f_\ast,g_\ast$ when $f\simeq g$? What about $f_!,g_!$ and $f^!,g^!$?

REPLY [2 votes]: By naturality, it suffices to consider the projection map $p : X \times I \to X$.  If $X$ and $I$ are locally conctractible Hausdorff spaces and $\mathcal{F}$ is a sheaf of abelian groups on $X$, then the Vietoris-Begle theorem (Bredon, Sheaf theory, II.13) states that the unit $\mathcal{F} \to p_* p^* \mathcal{F}$ is an isomorphism. There are also cohomological and Ext-functor versions of this result. See here in section 2.d.