In singularity theory, one defines an intrinsic derivative for a vector bunle homomorphism $\phi: E \rightarrow F$ where $E\xrightarrow{\pi_E}B$ and $F\xrightarrow{\pi_F}B$ are vector bundles with fibers diffeomorphic to $L'$ and $L$. This construction is useful for example to define the Hessian of a real valued map on a manifold at a critical point without an inner product structure present. This is my understanding of how it is defined:
For $b \in B$ and a small neighbourhood $U$ around $b$ in $B$, take trivializations for $\Phi$ and $\Psi$ for $E$ and $F$. In trivialized coordinates, $\phi$ could be thought of as a map $\phi:U\times L_1 \rightarrow U\times L_2$ given by $\phi(b, v) = (b, K(b, v))$ with $v \in L$, $K(b, v) \in L'$ and $K$ linear in $v$. Therefore, $DK(b, v)$ could be thoguht of as a bi-linear map of the form $T_bB \times E_b \rightarrow F_b$ where $E_b = \pi_E^{-1}(b)$ and $F_b = \pi_F^{-1}(b)$. Intrinsically however, this would define a coordinate independent bi-linear map, for each $b \in B$ of the form $T_bB \times \ker \phi_b \rightarrow \mathrm{coker} \phi_b$.
Question: I would like to see if this intrinsic derivative could be defined without coordinates. Since the intuition is that we are taking partial derivative with respect to $b$ for a smooth family of linear maps $\phi_b:E_b\rightarrow F_b, \quad \phi_b = \left.\phi\right|_{E_b}$, to get a bi-linear map, I suspect that we can use the same idea with which we define covariant derivative in the presence of connections. Especially that one could write a canonical isomorphism $T_{0(E_b)}E \cong \ker D\pi_E(0(E_b)) \oplus D0(b)(T_bB)$ and a similar decomposition for $F$. (Here $0$ is the zero section $B\rightarrow E$.)
