Let $f:X \to Y$ be continuous map between topological spaces and $E$ a $n$-vector bundle over $Y$. Then we know that the pullback $f^*E$ is also a $n$-bundle over $B$ and a pullback in sense of category theory.
Let assume that we have following commutative square
$$ \require{AMScd} \begin{CD} F @>{g} >> E \\ @VVpV @VVpV \\ X @>{f}>> Y \end{CD} $$
with another $n$-bundle $F$ and $g$ a bundle map. Since $f^*E$ is a pullback we obtain a continuous map $h:F \to f^*E$ of vector bundles.
My question is when is $h$ an isomorphism of vector bundles?
Does it suffice that $h$ induce a linear isomorphism $h_x: F_x \to f^*E_x$ on each fiber?
My considerations:
According to definition a morphism of vector bundles is an isomorphism iff
(1) it is a homeomorphism
(2) fiberwise a linear isomorphism
$h$ is by construction continuous and bijective. (2) holds by assumption.
Is then $h$ already an isomorphism or are there some pathological conterexamples? The problem is that I can't find an argument that the inverse map is also continuous.
Or does it suffice to verify that the map is local homeomorphism using trivialisations?