Let $f\colon M\to N$ a smooth surjective map of compact oriented manifolds of the same dimension. Then there is a map $f_!\colon H_i(N)\to H_i(M)$ obtained from the induced map on cohomology combined with Poincaré duality. This map has several names. I have seen it called the transfer, umkehr or wrong-way map. And it is treated in several question on mathoverflow and math.stackexchange:

Reference for push-pull formula in cohomology

I am interested in a geometric interpretation of this map in the following situation. Assume that there is an open subset $U\subset N$ such that, letting $V=f^{-1}(U)$, the restriction $f|_V\colon V\to U$ is an $r$-sheeted covering map ($r\in\mathbb{N}$) and $N'\subset U$ a compact oriented submanifold. If I understood correctly the answers and comments from the questions above, then we have

$f_!([N'])=[f^{-1}(N')]$.

Is this correct? And is there a reference for this? Several books in the answers and comments to the questions above are mentioned but I have actually not found this statement.

I should also say that I want to use this statement in a paper (if true!) and neither me nor the audience of the paper are experts on algebraic topology. So I would very much appreciate a reference where this fact is explicitly stated or at least a statement which easily implies this fact.

Edit: One can construct $f_!$ also in the case when $M$ and $N$ are oriented manifolds with boundary and $f^{-1}(\partial N)=\partial M$ (and all other assumptions the same). Now the map map $f_!\colon H_i(N)\to H_i(M)$ is obtained from the induced map on cohomology relative to the boundaries combined with Lefschetz duality. Is the conclusion $f_!([N'])=[f^{-1}(N')]$ still true? We still assume $f|_V\colon V\to U$ being a $r$-sheeted covering, neither $U$ nor $V$ intersect the boundaries of $M$ and $N$, and $N'\subset U$ a compact oriented submanifold (without boundary).