$ E \times_M E $ via the linear structure on $ E $, that is, $$\mathrm{vl}: E \times_M E \to \VBundle E, \quad (e, v) \mapsto \difFracAt{}{\varepsilon}{0} (e + \varepsilon v) \in \VBundle_e E.$$ The kernel $ \HBundle E \equiv \ker K $ of $ K $ is then a vector subbundle of $ \TBundle E $ such that $ \TBundle E = \HBundle E \oplus \VBundle E $ is a fiberwise topological isomorphism. Accordingly, the connection $ K $ yields a bundle isomorphism $$\label{eq:connection:decomposition}
    \TBundle E \to E \times_M \TBundle M \times_M E, \quad (Z_e) \mapsto \bigl(e, \tangent_e \pi(Z_e), K(Z_e)\bigr).$$ The inverse is given by $$\label{eq:connection:decompositionInverse}
    E \times_M \TBundle M \times_M E \to \TBundle E, \quad (e, X, v) \mapsto \mathrm{vl}(e, v) + X^K_e,$$ where $ X^K_e $ denotes the $ K $-horizontal lift of $ X $ to $ \TBundle_e E $. Given a smooth section $ \phi $ of $ E $, its covariant derivative $ \nabla \phi $ is defined by $$\label{eq:connection:covDeriv}
    \nabla_X \phi = K \bigl(\tangent \phi (X)\bigr)$$ for every $ X \in \VectorFieldSpace(M) $. Using the Leibniz rule, $ \nabla $ is extended to the exterior covariant derivative $ \dif_K: \DiffFormSpace^k(M, E) \to \DiffFormSpace^{k+1}(M, 