I have a question about the explanation of the data defining a so-called equivariant sheaf $ F $ on an X wiki schema: https://en.wikipedia.org/wiki/Equivariant_sheaf. Let denote by $ sigma: G times_S X to X $ an action of a group $ G $ sure $ X $ . Then a $ O_X $-module $ F $ is called equivariant if it exists in isomorphism $ phi: sigma ^ * F simeq p_2 ^ * F $ of $ mathcal {O} _ {G times_S X} $-modules and in addition the condition "cocycle" $ p_ {23} ^ * phi circ (1_G times sigma) ^ * phi = (m times 1_X) ^ * phi $ is satisfied where $ p_ {23}, 1_G times sigma, m times 1_X $ a map between $ G times G times X $ and $ G times X $.

FOLLOWING EXPLANATION THAT I DO NOT UNDERSTAND: Then, we say that the condition of cocycle indicates that at the level of the stems, the isomorphism $ F_ {gh cdot x} simeq F_x $ is the same as $ F_ {g cdot h cdot x} simeq F_ {h cdot x} simeq F_x $; that is, it reflects associativity. (*)

What I do not understand is why induced isomorphism $ (m times 1_X) ^ * phi $ provides a map $ F_ {gh cdot x} to F_x $ at the level of the stems?

To know I do not see why $ F_ {gh cdot x} $ and $ F_x $ are the domain and the correct codomaine of this map $ (m times 1_X) ^ * phi $.

Indeed, the induced isomorphism $ (m times 1_X) ^ * phi $ is an iso of sheaf $$ (m times 1_X) ^ * sigma ^ * F = ( sigma circ (m times 1_X)) ^ * F to (m times 1_X) ^ * p_2 ^ * F = (p_2 circ (m times 1_X)) ^ * F $$.

Fix a point $ (g, h, x) in G times G times X $.

then $ sigma circ (m times 1_X) (g, h, x) = gh cdot x $ and $ p_2 circ (m times 1_X) (g, h, x) = x $

I do not understand why the stems of $ (g, h, x) $ domain and codomaine are given by $$ (( sigma circ (m times 1_X)) ^ * F) _ {(g, h, x)} = F_ {gh cdot x} $$ and $$ ((p_2 circ (m times 1_X)) ^ * F) _ {(g, h, x)} = F_x $$ as shown in (*)?

I've learned that generally for a morphism $ f: X to Y $ and a sheaf $ F $ sure $ Y $ we have the following formula for the stem in $ z in X $ of the sheaf of draw:

$$ (f ^ * F) _z = O_ {X, z} otimes_ {O_ {Y, f (z)}} F_ {f (z)} $$.

Now, we apply this formula to our situation, so $ f = sigma circ (m times 1_X) $ and $ z = (g, h, z) $ so we get

$$ (( sigma circ (m times 1_X)) ^ * F) _ {(g, h, x)} = O_ {G times G times X, (g, h, x)} otimes_ {O_ {X, gh cdot x}} F_ {gh cdot x} $$

But (*) says that it should be equal to $ F_ {gh cdot x} $. Why? Where is the error in my reasoning? What's going on with the left "factor"?