Cocycle condition for transition functions

Let $\{(U_i,\Phi_i)\}$ be a bundle atlas for a smooth fiber bundle with typical fiber $F$ , and let $t_{ij}:U_{ij}\to \mathrm{Diff}(F)$ be the associated transition functions. They satisfy the following identities:

Identity on the diagonal: $t_{ii}(x)=\mathrm{id}_F$ for all $x\in U_i$ .
Inverse on overlaps: on $U_{ij}$ , one has $t_{ji}(x)=t_{ij}(x)^{-1}$ .
Cocycle condition on triple overlaps: on $U_{ijk}=U_i\cap U_j\cap U_k$ , $t_{ij}(x)\circ t_{jk}(x)=t_{ik}(x)\qquad \text{for all }x\in U_{ijk}.$

These conditions are exactly the statement that the changes of trivialization compose consistently, so that the local products $U_i\times F$ glue to a well-defined smooth fiber bundle.

Examples

Trivial bundle: all $t_{ij}$ are the identity, so the cocycle condition holds tautologically.
Möbius line bundle: cover the circle by two arcs whose overlap has two components. Choose $t_{12}=+1$ on one component and $t_{12}=-1$ on the other, with $t_{21}=t_{12}^{-1}$ . These satisfy the cocycle identities and give the nontrivial line bundle. Choosing $t_{12}\equiv-1$ on the entire overlap instead gives a trivial bundle after reversing one local frame.
Tangent bundle: on triple overlaps of coordinate charts, the cocycle condition is the chain rule for Jacobians of coordinate changes.