Covariant derivative of vector bundle map $\Psi:\xi\rightarrow \eta$

Question: "I guess one idea I had was to consider Ψ as an element of the endomorphism bundle End(ξ,η) and then perhaps there is an induced connection on End(ξ,η) from the ones in η and ξ such that ∇ηsΨX=(∇End(ξ,η)sΨ)X+Ψ(∇ξsX). Is this going to be the case ? Any help is appreciated, thanks in advance."

Answer: If $E,F$ are finite rank vector bundles with connections $\nabla_E, \nabla_F$, there is a "canonical" connection $\nabla$ on $Hom(E,F) \cong E^*\otimes F$ - the tensor product of $\nabla_E$ and $\nabla_F$. Given any $\phi \in Hom(E,F)$ and any vector field $x$ you have thus a definition of $\nabla(x)(\phi)$. It has the property that for any section $s$ it follows

$$\nabla(x)(s\phi)=s\nabla(x)(\phi)+x(s)\phi.$$

You may define

$$\nabla(x)(\phi):= \nabla_F(x) \circ \phi - \phi \circ \nabla_E.$$