In singular (co)homology, if $\alpha\in C^*(X)$ and $x\in C_*(X)$, then the cap product $\alpha \cap x$ is generally defined by the following process:

Apply to $x$ the diagonal map $C_*(X)\to C_*(X\times X)$ followed by some choice of Alexander-Whitney chain equivalence $ C_*(X\times X)\to C_*(X)\otimes C_*(X)$ to obtain an element $\sum y_i\otimes z_i$.

Apply $\alpha$ to $\sum y_i\otimes z_i$ by the slant product, or, in other words and roughly speaking, apply $\alpha$ to ``half of the factors''.

Depending on your favorite conventions (and what you're trying to accomplish), there may also be a sign (which might also be part of your definition of the slant product - but let's ignore this).

The reason I'm being so cagey with wording in part 2 is directly related to my question: In almost all major textbook sources I have consulted, step 2 is performed by forming $\sum y_i \alpha(z_i)$, which strikes me as somewhat unnatural, forcing the $\alpha$, which starts off on the left to jump all the way over the $y_i$ terms to get to the $z_i$ terms on the right. Is there a good mathematical reason for this convention? Why not define the cap product to be $\sum \alpha(y_i) z_i$?

The one major exception to this convention seems to be Hatcher. He does form $\sum \alpha(y_i) z_i$, but he also writes cap products as $x\cap \alpha$, so his cochain also has to jump, but it jumps over the $z$s instead!

(For the record, I'm not asking this question out of idle pickiness. Jim McClure and I have been doing a lot of work with cap products recently, and we're trying to be consistent amongst various conventions for various issues, but preferably with good reasons thrown in!)