Let $(E,\mathcal{O})$ be an elliptic curve over a (perfect) field $K$ and let $\textbf{Cov}(E,\mathcal{O})$ denote the category of finite pointed étale covers of $E$ from smooth varieties where the distinguished point $\mathcal{O}'$ on each cover is a $K$-point. One can show that each of these covering varieties must be an elliptic curve $(E', \mathcal{O}')$ and the covering maps are then isogenies of elliptic curves. The morphisms $(E', \phi)\rightarrow (E'', \psi)$ in this category between two finite étale covers are pointed morphisms of algebraic varieties, and thus become finite étale isogenies in their own right (since the "base change" of an étale morphism is étale).
Let $\textbf{Cov}_{\mathbb{N}} (E,\mathcal{O})\subseteq\textbf{Cov}(E,\mathcal{O})$ denote the full subcategory consisting of multiplication-by-$m$ isogenies $[m]:E\rightarrow E$ for $m\in K^\times$ (see this other question of mine). I'm wondering if there is a functor $\Delta: \textbf{Cov}(E,\mathcal{O})\rightarrow\textbf{Cov}_{\mathbb{N}}(E,\mathcal{O})$ sending each isogeny $\phi: E'\rightarrow E$ to the precomposition with the dual isogeny $[\deg\phi] = \phi\circ\hat{\phi}:E\rightarrow E$?
More precisely, if $f: (E', \phi)\rightarrow(E'', \psi)$ is a morphism in $\textbf{Cov}(E,\mathcal{O})$, does there exist a (functorial) morphism $\Delta(f):E\rightarrow E$ in $\textbf{Cov}_{\mathbb{N}}(E,\mathcal{O})$ such that the following diagram commutes?