Let $X\xrightarrow{f} Y \xrightarrow{g} Z$ be a sequence of smooth maps of manifolds, and assume that $g$ is transverse to a submanifold $W$ of $Z$. Show $f\pitchfork g^{-1}(W) \leftrightarrow g\circ f \pitchfork W$.
From the definition of transverse, $$\forall x \in g^{-1}(W), \text{im}d_xg + T_ZW = T_ZZ$$
Then I was trying to start the first side of the proof, $f\pitchfork g^{-1}(W) \rightarrow g\circ f \pitchfork W$, but I'm confused about $f \pitchfork g^{-1}(W)$. Does it mean $$\forall x \in W, \text{im}d_xf + T_ZW = T_ZZ$$
I'm not sure where to go from there either.