In May and Sigurdsson's *Parameterized Homotopy Theory*, Proposition 2.2.11, four isomorphisms of functors are given. For a pullback square of base spaces $C=holim(A\overset{f}\to B\overset{j}\leftarrow D)$, with induced maps $C\overset{g}\to D$ and $C\overset{i}\to A$, there are natural isomorphisms:


$$(1)~~j^\ast f\_!\cong g\_!i^\ast~~~(2)~~f^\ast j\_\ast \cong i\_\ast g^\ast~~(3)~~f^\ast j\_!\cong i\_!g^\ast~~~(4)~~ j^\*f\_\ast\cong g\_\*i^\*$$


Where for a map $f$ of base spaces, $f^\ast$ is the usual pullback of a parameterized **space**, $f\_\ast$ is its right adjoint, and $f\_!$ is its left adjoint. 


Do these same natural equivalences hold for parameterized **spectra** over a pullback diagram of spaces? In particular, I am interested in equivalence $(3)$ above. And even more specifically, I am interested in comparing the two sides when $A$ above is a contractible space. In other words, $C$ is the homotopy fiber of the map $D\to B$. In that case, for a bundle of sphere spectra over $B$, we have that $i\_!g^\ast$ yields the Thom spectrum of the pullback. How then should we think of $f^\ast j\_!$ of the same bundle of sphere spectra?



Shriek push-forward for parameterized spectra
Yes, the [Beck-Chevalley condition](http://ncatlab.org/nlab/show/Beck-Chevalley%20condition) holds for [parameterized spectra](http://ncatlab.org/nlab/show/parametrized+spectrum). A proof appears as prop.4.3.3 in 


* Michael Hopkins, Jacob Lurie, *Ambidexterity in K(n)-Local Stable Homotopy Theory* ([pdf](http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf), [nLab](http://ncatlab.org/nlab/show/Ambidexterity+in+K%28n%29-Local+Stable+Homotopy+Theory)).


Some chat related to this is also around example 5.6 in my note "Quantization via Linear homotopy types" ([arXiv:1402.7041](http://arxiv.org/abs/1402.7041))


