I am working on understanding Peter May's "A genaral algebraich approach to Steenrod operations", so for this purpose I am trying to compare his framework with explicit construction of Steenrod squares, as described in Hatcher's "Algebraic Topology", Section 4.L, p. 502.
So the construction should go as follows: Let $p=2$, $X$ be a topological space. We are taking all cochains and cohomology with coefficients in $\mathbb{Z}/2$. We have a following sequence of maps
$$ *\times X\xrightarrow{i}\mathbb{S}^\infty\times X\xrightarrow{\Delta}\mathbb{S}^\infty\times X^2, $$ where $i$ is given by the inclusion of the basepoint and $\Delta$ is a diagonal map. This maps induce on the cochain level a following map $$ \theta :C^*(\mathbb{S}^\infty)\otimes C^*(X)^{\otimes 2}\to C^*(X). $$ We note that $\mathbb{S}^\infty$ is a model for $E\Sigma_2$, so by putting $W_{-q}=C^q(\mathbb{S}^\infty)$ we get a standard $\Sigma_2$-resolution of $\mathbb{Z}/2$. So we are in the situation required by Def.2.1. of "A general algebraic approach..." and the conditions i)-ii) are satisfied.
Note that $\theta$ induces a map $$ \tilde{\theta}:W\otimes_{\Sigma_2}C^*(X)^{\otimes 2}\to C^*(X). $$
Now let $x\in H^q(X)$ and $e_{-i}\in C^i(\mathbb{S}^\infty )$ be a generator. We define $D_i(x)\in H^{2q-i}(X)$ by $D_i(x)=\tilde{\theta}^* (e_{-i}\otimes x^2)$, where $\tilde{\theta}^*: H(W\otimes _{\Sigma_2}C^*(X)^{\otimes 2})\to H^*(X)$.
Thus we define $Sq^k:H^q(X)\to H^{q+k}(X)$ by $Sq^k(x)=D_{q-k}(x)$, where $D_i=0$ if $i<0$.
In order to get the same result as in the Hatcher's construction, we observe that $\tilde{\theta}^*$ is induced by $\Delta^*i^*$, so it factors through $H(W\otimes_{\Sigma_2} C^*(X))\cong H^*(\mathbb{R}P^\infty)\otimes H^*(X)$, where the isomorphism is given by Eilenberg-Zilber theorem and the Kunneth theorem. The image of $e_i\otimes x^2$ by $\Delta^*$ under this isomorphism may be written as $\Sigma_{i}t^{q-i}\times a^i$, and then we obtain a $Sq^k(x)$ as $i^*(t^{q-k}\times a^k)$.
Is that construction correct? And does it fit to the Peter May's framework?
