Local connection forms glue exactly when they obey the transformation law

Necessity is the transformation formula for each pair of frames. For sufficiency, write $s=e_{\alpha }{u}_{\alpha }$ on $U_{\alpha }$ and define there ${\eta }_s=e_{\alpha }(d{u}_{\alpha }+{\omega }_{\alpha }{u}_{\alpha })$. This is a smooth Hom section since its local entries are sums and products of smooth functions and their derivatives.

On an overlap put $A=A_{\alpha \beta }$, so $u_{\alpha }=A{u}_{\beta }$. The ordinary product rule and the assumed matrix identity give $d{u}_{\alpha }+{\omega }_{\alpha }{u}_{\alpha }=Ad{u}_{\beta }+(dA+{\omega }_{\alpha }A)u_{\beta }=A(d{u}_{\beta }+{\omega }_{\beta }{u}_{\beta })$. Multiplying by $e_{\alpha }$ shows the two definitions agree. They therefore assign a unique global value at every point; smoothness holds because near each point it is one of the displayed smooth local expressions.

The local rule is real-linear, and $d(fu)+\omega (fu)=dfu+f(du+\omega u)$ proves its Leibniz identity. Consequently $s\mapsto {\eta }_s$ is a connection. Any connection with the prescribed matrices must have these local values, by its Leibniz rule applied to each frame expansion, proving uniqueness. Empty overlaps require no matching; an empty base has the unique zero connection. A single chart needs no gluing, while rank zero and base dimension zero make the expression zero. The agreed values in step 2.1 specify the global object uniquely without choosing a chart for every point.