Are the connecting homomorphism induced by Kummer sequence and that of localization sequence commutative?

In other words, is the following statement true?

If it is true, then, how can one prove it?

Please give me any advice.

STATEMENT: Let $X$ be a scheme, $n$ an integer invertible on $X$, $Z$ a closed subscheme of $X$ and $U=X\setminus Z$.

Let $G_{m}$ be an etale sheaf on $X$ defined by sections invertible.

And let $\mu_{X}$ be an etale sheaf on $X$ defined by sections which are n-th roots of 1.

Then, the following two homomorphisms are the same.(Sorry, I don't know how to draw a diagram...)

$H^{i}(U, G_{m}) \stackrel{localization}{\to} H_{Z}^{i+1}(X, G_{m}) \stackrel{Kummer}{\to} H_{Z}^{i+2}(X, \mu_{X})$,

$H^{i}(U, G_{m}) \stackrel{Kummer}{\to} H^{i+1}(U, \mu_{X}) \stackrel{localization} \to H_{Z}^{i+2}(X, \mu_{X})$.

Here, morphisms are connectiong homomorphisms of long exact sequences.