Let $\pi:E\to N$ be a smooth fibre bundle with typical fibre $F$, and let $f:M\to N$ be smooth. Then $f^*E$ has a unique smooth fibre-bundle structure over $M$ for which the canonical map $\tilde f:f^*E\to E$ defined by $\tilde f(x,e)=e$ is smooth and each local product chart of $E$ over $V\subset N$ induces a local product chart of $f^*E$ over $f^{-1}(V)$.