What is the essence of ``the naturality of the cap product''? Associated to a continuous map $f : X → Y$, there are natural pushforward and pullback maps on homology and cohomology, respectively, denoted $f_∗ : H_∗(X) → H_∗(Y)$ and $f^*: H^* (Y) → H^* (X)$.
These are related by the projection formula, also called ``the naturality of the cap product'':
$$f_∗(f^∗ c \cap σ) = c \cap f_∗ σ.$$
My questions are that:

*

*what is the essence of this ``the naturality of the cap product''?


*what are the uses of this ``the naturality of the cap product''?
Suppose it can be used in the proof of Alexander-Lefschetz-Poincaré Duality. What are the essence behind?
 A: *

*One way to think about the cap product, which is a pairing
$$ \cap\colon H_{k+l}(X)\otimes H^k(X) \rightarrow H_l(X) $$
is via its adjoint which is a homomorphism
$$ \alpha_X \colon H_{k+l}(X)\rightarrow \hom(H^k(X),H_l(X)). $$
Notice that both sides of the equation are covariant functors in $X$ and naturality of the cup product precisely means that $\alpha_X$ is a natural transformation.


*A very important application of this naturality statement is the following: Let
$$f\colon M \rightarrow N$$ be a continous map of degree $d$ between closed, connected and oriented manifolds of dimension $n$. The Poincaré duality isomorphism is defined as the "capping with the fundamental class of a manifold":
$$ PD_M = \_\cap [M]\colon H^k(M) \rightarrow H_{n-k}(M).$$
Let us apply naturality of the cap product to our map $f$.
For $\phi\in H^k(N)$:
$$f_*(f^*(\phi)\cap [M]) = \phi \cap d\cdot [N] =d\cdot (\phi\cap [N]). $$
If $f^!\colon H_k(N)\rightarrow H_k(M)$ denotes the "Umkehr map" $PD_M\circ f^* \circ PD^{-1}_N$, the the above shows that $f_* \circ f^! \colon H_k(N)\rightarrow H_k(N)$ is multiplication by $d$, which can be useful. In particular, if $d=1$, then $f_*$ is split surjective. Also, using the Theorem of Whitehead, a map of degree $1$ from a sphere to a closed and simply connected manifold is a homotopy equivalence.
