Vector-bundle pullback is canonically functorial

The maps $(z,e)\mapsto e$ and $e\mapsto (q(e),e)$ are mutually inverse fiberwise-linear continuous maps ${\mathrm{id}}_Z^{\ast }E\rightleftarrows E$. They are continuous by the product and subspace formulas in [F1].

The composite comparison is $\alpha (x,(f(x),e))=(x,e)$, with inverse $(x,e)\mapsto (x,(f(x),e))$. Both maps are continuous restrictions of product-coordinate maps, are linear on each fiber, and preserve the defining equations, so [F1] makes them inverse bundle isomorphisms.

For a third base map, every route through the associativity comparisons deletes the same redundant base coordinates and ends at the same pair $(w,e)$. For a bundle map, both naturality routes apply that map to the same final $e$-coordinate. Thus the comparisons are coherent and natural.

The pullback section formula sends $x$ to $(x,s(g(f(x))))$ whether it is applied once along $gf$ or twice along $g$ and $f$; the identity formula similarly reduces to $s$. Hence sections respect identities and composition under the canonical comparisons.