A canonical equivariant structure on the structure sheaf: checking the cocycle condition Let $G$ be a linear algebraic group, let $X$ be a $G$-variety (for simplicity, let $X$ be a complex affine variety, with associated structure sheaf $\mathcal O_X$, regarded as a sheaf of $\mathcal O_X$-modules) with $G$-action $\sigma: G \times X \to X$ and projection $p: G \times X \to X$. Then $\sigma^* \mathcal O_X$ and $p^* \mathcal O_X$ are sheaves on $G \times X$ (considered as the product of two varieties).
Since $$ \sigma^* \mathcal O_X = \mathcal O_{G \times X} \otimes_{\sigma^{-1} \mathcal O_X} \sigma^{-1} \mathcal O_X \cong \mathcal O_{G \times X} $$ and a similar expression holds for $p^* \mathcal O_X$, there is a natural isomorphism $\sigma^* \mathcal O_X \cong p^* \mathcal O_X$.
I would like to show that this satisfies the cocycle condition in the definition of $G$-equivariant sheaves, namely
$$ p_{23}^* \phi \circ (1_G \times \sigma)^* \phi = (m \times 1_X)^* \phi,$$
where $p_{23} : G \times G \times X \to G \times X$ is projection along the first factor and $m$ is the group multiplication on $G$.
Here is what I have tried.
Evaluate (I think it makes sense to 'evaluate' these morphisms, considered as morphisms on $\text{Sh}(G \times G \times X)$) the left-hand side on the sheaf $(1_G \times \sigma)^* \sigma^* \mathcal O_X$:
$$
\begin{aligned} p_{23}^* \phi \circ (1_G \times \sigma)^* \phi \left( (1_G \times \sigma)^* \sigma^* \mathcal O_X \right)
&= p_{23}^* \phi \circ (1_G \times \sigma)^* \left( \phi (\sigma^* \mathcal O_X) \right) \\
&= p_{23}^* \phi \circ (1_G \times \sigma)^* \left( p^* \mathcal O_X \right) \\
&= p_{23}^* \phi \circ (1_G \times \sigma)^* \mathcal O_{G \times X} \\
&= p_{23}^* \phi \left( \mathcal O_{G \times G \times X} \otimes_{(1_G \times \sigma)^{-1} \mathcal O_{G \times X}} (1_G \times \sigma)^{-1} \mathcal O_{G \times X} \right) \\
&= p_{23}^* \phi (\mathcal O_{G \times G \times X}). \end{aligned} $$
The first equality I justify since pullback is functorial, and the rest is just unpacking definitions.
What I am confused on is how to continue - the thing I have ended up with seems resistant to further manipulation. Moreover, evaluation of the right-hand side also confuses me - the expression $$ (m \times 1_X)^* \phi \left( (1_G \times \sigma)^* \sigma^* \mathcal O_X \right)$$ looks like nonsense to me.
I understand vaguely how the cocycle condition reflects the group associativity, though this I am still not sure on (I have the example of $G = \mathbb{C}^\times, X = \mathbb{V}(x)$ in mind), but I do not see how to concretely show $\mathcal O_X$ is an equivariant sheaf.
What am I missing in all of this?
 A: The calculation is very straightforward once it is observed that there are three commutative diagrams projecting from $G \times G \times X$ to $X$:
$$ 
\require{AMScd}
\begin{CD}
G \times G \times X   @>m \times 1>>   G \times X\\
@VV1 \times aV                                    @VVaV\\
G \times X                 @>a>>      X
\end{CD},
\require{AMScd}
\begin{CD}
G \times G \times X   @>p_{23}>>   G \times X\\
@VV1 \times aV                                    @VVaV\\
G \times X                    @>p>>      X
\end{CD},
\require{AMScd}
\begin{CD}
G \times G \times X   @>p_{23}>>   G \times X\\
@VVm \times 1V                                    @VVpV\\
G \times X                    @>p>>      X
\end{CD}.
$$
In this way, there are three sets of equalities that hold (regardless of if $\mathcal F$ is equivariant or not):
$$
\begin{aligned}
(1 \times a)^* a^* \mathcal F &= (m \times 1)^* a^* \mathcal F, \\
(1 \times a)^* p^* \mathcal F &= p_{23}^* a^* \mathcal F, \\
(m \times 1)^* p^* \mathcal F &= p_{23}^* p^* \mathcal F. \\
\end{aligned}
$$
Using these three equalities, the calculation can be fixed.
