Let $ f:X \rightarrow Y$ be finite covering map of simplicial sets of finite degree, say $d$. Let $\varSigma^{\infty}$ denote the functor from the category of simplicial sets to spectra which is an additive category, where


$\varSigma^{\infty}Y = \{n \mapsto S^{n} \wedge Y \}$. 


I want to know the sketch or reference for the proof of the following statement:"$f$ induces a transfer map $ f': $ $\varSigma^{\infty}Y \rightarrow \varSigma^{\infty}X $ such that $f'' \cdot f'$ is multiplication by $d$ in the abelian group $Hom(\varSigma^{\infty}Y, \varSigma^{\infty}Y )$. Note $f''$ is the obvious map $\varSigma^{\infty}X \rightarrow \varSigma^{\infty}Y $ induced by functor $\varSigma^{\infty}$.


"



Transfer map of simplicial sets
In the more general version with compact (finitely dominated) fibers, this is called the **Becker-Gottlieb transfer**. You can find a long list of references on the [nlab](https://ncatlab.org/nlab/show/Becker-Gottlieb+transfer). Here are a few of them:


* *Becker, James C.; Gottlieb, Daniel H.*, [**The transfer map and fiber bundles**](http://dx.doi.org/10.1016/0040-9383(75)90029-4), Topology 14, 1-12 (1975). [ZBL0306.55017](https://zbmath.org/?q=an:0306.55017).
* *Becker, James C.; Gottlieb, Daniel H.*, [**Vector fields and transfers**](http://dx.doi.org/10.1007/BF02568269), Manuscr. Math. 72, No.2, 111-130 (1991). [ZBL0736.55012](https://zbmath.org/?q=an:0736.55012).
* *Klein, John; Malkiewich, Cary*, [**The transfer is functorial**](https://arxiv.org/abs/1603.01872), arXiv:1603.01872.


In particular, for the result you are looking for see the footnote at page 5 of the last reference.


