Let $F$ be a finite field, $Sq^i$ be the $i$-th Steenrod operation

$$ H^*(-;F) \to H^{*+i}(-;F).$$

By Yoneda lemma, such operation is a map $\phi_i: B^{*+i}F \to B^* F$, where $B$ denotes the delooping operator. By applying its inverse $\Omega$ many times, we get a map $\psi_i: F \to \Omega^iF$. I would like to understand Steenrod operation $Sq^i$ by $\psi_i$.

1. Can you describe $\psi_i$ explicitly? Is it just the canonical inclusion?
2. What do Adem's relations translate to in this perspective? Can I see the combinatorics much clearly?
3. Can we generalize this construction from $F$ to all abelian groups? It seems to me that the crux hides in the natural transformation

$$Id \to \Omega^i,$$

and does nothing with the coefficient system $F$.

Let $F$ be a finite field, $Sq^i$ be the $i$-th Steenrod operation

$$ H^*(-;F) \to H^{*+i}(-;F).$$

By Yoneda lemma, such operation is a map $\phi_i: B^{*}F \to B^{*+i} F$, where $B$ denotes the delooping operator. By applying its inverse $\Omega$ many times, we get a map $\psi_i: \Omega^iF \to F$. I would like to understand Steenrod operation $Sq^i$ by $\psi_i$.

1. Can you describe $\psi_i$ explicitly?
2. What do Adem's relations translate to in this perspective? Can I see the combinatorics much clearly?
3. Can we generalize this construction from $F$ to all abelian groups? It seems to me that the crux hides in the natural transformation

$$\Omega^i \to Id,$$

and does nothing with the coefficient system $F$.