I am reading the Transversality Theorem and I had to remind myself what the strange definitions mean.
For manifolds,
Let $X,Y \subset Z$ be sub manifolds and $x \in X \cap Y$, then $X \pitchfork Y $ means their tangent space $T_x X + T_x Y = T_x Z$
For maps
Let $f: X \to Y$ be a smooth map and $Z \subset Y$ be a sub manifold. Then $f$ is traverse to $Z$ if $$Im(df_x) + T_{f(x)}Z = T_{f(x)}Y$$
So here is what my definition of what I thought $f \pitchfork Z$ would mean. If given a smooth map $f: X \to Y$ and sub manifold $Z \subset Y$. Then to suggest a definition for $f \pitchfork Z$, one would consider
Let $a \in f(X) \cap Z$ (so there is an assumption that $f(X) \subset Y$ is already a sub manifold), then $f(X) \pitchfork Z$ means $$T_af(X) + T_aZ = T_aY$$
Let $\phi: U \to R^{\dim f(X)}$ be a local parametrization where $U\subset R^k$, then we define $$T_af(X) = Im(\{d\phi_0 : d\phi_0 : R^k \to R^{\dim f(X)} \}).$$ Now I am almost certain what am I writing is the same because there is an isormophism $T_x(X) \approx R^k$, so $df_x$ and $d\phi_0$ must be the same up to diffeomorphism is what I am suspecting. But I am not sure. I am guessing transversality between maps and sub manifolds gives more freedom of thought in geometry?