Below are two different stories about power operations for $\mathbb{E}_\infty$-ring spectra, and I am struggling to see how they relate. In the following we let $R$ be an $\mathbb{E}_\infty$-ring spectrum.

- $R$ admits natural maps $R^{\wedge n}_{h\Sigma_n} \to R$. If $X$ is now a space, we may apply $(\,\cdot\,)^{\wedge n}_{h\Sigma_n}$ to a $\Sigma^\infty_+ X \to R$ representing an element of $R^0(X)$, which we then compose with the multiplications $R^{\wedge n}_{h\Sigma_n} \to R$ to obtain a map $$\mathbb{P}_n \colon R^0(X) \to R^0(X^{\times n}_{h\Sigma_n})$$ called the $n$-th total power operation of $R$ --- a multiplicative but non-additive map.
- There is a category $\mathsf{CAlg}(R)$ of $R$-algebras, and it admits a forgetful functor $U \colon \mathsf{CAlg}(R) \to \mathsf{Sp}$. One then defines a spectrum of power operations on $R$ to be the endomorphism spectrum $\operatorname{Map}(U,U)$.

Question.Are these two approaches in any way related?

At first glance it appears not so, but the forgetful functor $U$ admits a left adjoint $F$ sending a spectrum $Y$ to $R \wedge \bigoplus_n Y^{\wedge n}_{h\Sigma_n}$ --- a formula vaguely similar to what we see in the first definition. I guess if you apply the second definition to the $R$-algebra $\operatorname{Map}(\Sigma^\infty_+ X,R)$ and play around with the adjunction you can make the comparison precise, but I'm struggling with the details.