This is a crosspost of [this MSE question](https://math.stackexchange.com/questions/1351300/how-do-the-direct-and-inverse-image-sheaf-functors-interact-with-homotopy).


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).



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> 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 $f^!,g^!$?
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> 
> 



How do the direct and inverse image sheaf functors interact with homotopy?
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](http://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=8495) in section 2.d.


