Pullback connections intertwine parallel transport

The fibre identifications give the canonical isomorphism ${\gamma }^{\ast }(f^{\ast }E)\cong (f\circ \gamma {)}^{\ast }E$. Functoriality identifies their connections and hence their derivatives in direction ${\mathrm{\partial }}_t$. Thus they identify parallel sections, piecewise and continuously at all corners. Explicitly, both coefficient equations use ${\omega }_{f(\gamma (t))}(d{f}_{\gamma (t)}\dot{\gamma }(t))$.

A parallel section on the left with initial vector $v$ corresponds to one on the right with initial vector $J_{a}v$. Uniqueness and endpoint evaluation give the asserted identity on every $v$. Constant $f$ gives zero matrix in a fixed target fibre frame; no injectivity of $f$ or nonzero curve velocity is required. For $a=b$ the two sides are $J_a$, and rank-zero bundles have the unique maps. The identifications are canonical, not selected trivializations.