TITLE: Functoriality for wrong way maps
QUESTION [9 upvotes]: In the K-theory formulation of the index theorem one defines the topological index in terms of the so called wrong way maps. Those maps are defined for embeddings of compact manifolds $i:X \to Y$: see section 2.2 (page 16) in Landweber - K-theory and elliptic operators article for more details. I would like to understand 

why this construction is functorial, i.e. why $(j \circ i)_!=j_! \circ i_!$ where $i:X \to Y$ and $j:Y \to Z$ are embeddings of compact manifolds. 

Forgive me if this question is too elementary for this site.

REPLY [4 votes]: You will find a careful proof of this in Eldon Dyer's monograph "Cohomology theories". It is true for any multiplicative cohomology theory $h^*$ and smooth maps $i:X\to Y$ and $j:Y\to Z$ which are $h^*$-oriented, meaning with a choice of Thom class for their stable normal bundles.
The definition of the Umkehr is on page 53, and the functoriality result is stated as Theorem 8 on page 57.