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. 


1. $R$ 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.
2. 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 $\operatorname{Map}(U,U)$.



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> **Question.** Are these two approaches in any way related?
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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.



Two definitions of power operations --- how do they relate?
The first observation is that $U$ is representable in $CAlg(R)$ by $F(S)$ (with $F$ as in your question): $$Map\_{CAlg(R)}(F(S),A) \simeq Map\_{SMod}(S,A) \simeq U(A).$$


By some appropriately flowery version of Yoneda's Lemma, it follows that
$$ Hom(U,U) \simeq Map\_{CAlg(R)}(F(S),F(S)) \simeq Map\_{SMod}(S,F(S)) \simeq R \wedge \bigvee\_n B\Sigma\_{n+}.$$
Applying $\pi\_0$ to this, is easy to see that the operation corresponding to $1 \in R\_0(B\Sigma\_n)$ will be precisely the classic $n$th power operation, when applied to the $R$--algebra $A = Map(\Sigma^{\infty}\_+X,R)$.


**Added later by request ...**


Suppose given $f \in \pi\_0(F(R))$. Tracing through my equivalences, the associated operation $\theta\_f: \pi\_0(A) \rightarrow \pi\_0(A)$, for $A$ a commutative $R$--algebra, is as follows.


Firstly $f$ can regarded as an $R$-module map $f:R \rightarrow F(R)$. Similarly, $x \in \pi\_0(A)$ can be regarded as an $R$-module map $x:R \rightarrow A$. Then $\theta\_f(x)$ is the composite
$$ R \rightarrow F(R) \rightarrow F(A) \rightarrow A,$$
where the first map is $f$, the next is $F(x)$ and the the last
is the structure map for the $R$--algebra: the wedge over $n$ of the maps $A^{\wedge n}\_{h \Sigma\_n} \rightarrow A$.


For the $n$th power operation, recall that $\displaystyle F(R) = R \wedge \bigvee\_m B\Sigma\_{m+}$, and let $f\_n$ be the composite
$$S^0 \rightarrow B\Sigma\_{n+} \rightarrow R \wedge B\Sigma\_{n+}\hookrightarrow F(R).$$
Then $\theta\_{f\_n}$ will be the $n$th power operation. You can now specialize to $A = Map(X,R)$ if you want.


