I'm reading Dolgachev's book 'Lectures on invariant theory'. In Chapter 7, the linearization of a group action is discussed. Let $G$ be a linear algebraic group acting on a quasi-projective variety $X$ over an algebraically closed field $k$ via $\sigma: G\times X\rightarrow X$. He then defines the notion of a $G$-linearized line bundle $L$ on $X$, and comments that for each $g\in G$ and $x\in X$, the induced map on the fibers $L_x\rightarrow L_{gx}$ is a linear isomorphism. We can view the set of such isomorphisms as of line bundles $$\overline{\sigma}(g): L\rightarrow g^*L\ ,$$ which satisfy certain cocycle conditions.

Upto here it's fine. Next, he makes the following remark: the collection of isomorphisms $\overline{\sigma}(g)$ can also be viewed as an isomorphism of line bundles $$\Phi : p_2^*L\rightarrow \sigma^*L\ ,$$ where $p_2:G\times X\rightarrow X$ is the second projection.

This last part is not clear to me. Surely, given such a $\Phi$, we can find the $\overline{\sigma}(g)$'s by restricting. But how to go the other way? Meaning, why do the collection of isomorphisms $\overline{\sigma}(g)$ given 'fiberwise' glue to give a global isomorphism? Is there a general theme like this, i.e. to define a morphism of sheaves, it is enough to define it on fibers in certain cases?

Thanks in advance!