Let $X$ be a compact complex manifold and $E$ be a finite dimensional holomorphic vector bundle on $X$ with a fixed $\bar{
\partial}$-connection $\bar{\partial}_E$.

Now we consider a small neighborhood $\Delta$ of $0$ in $\mathbb{C}$. Let $\epsilon_1(t)$ and $\epsilon_2(t)$ be two deformations of $\bar{\partial}_E$, i.e. they satisfy

 1. $(\bar{\partial}_E+\epsilon_i(t))^2=0$, $i=1,2$.
 2. $\epsilon_i(t)$ is holomorphic with respect to $t$, i.e. $\epsilon_i(t)\in \Omega^{0,1}(X, \text{End}(E))\otimes \mathcal{O}(\Delta)$, $i=1,2$.
 3. $\epsilon_i(0)=0$,  $i=1,2$.

Let $\phi(t)\in C^{\infty}(X\times \Delta,\text{End}(E))$ be such that

 1. $\phi(t)\circ (\bar{\partial}_E+\epsilon_1(t)=(\bar{\partial}_E+\epsilon_2(t))\circ \phi(t)$.
 2. $\phi(0)\equiv \text{id}_E$ on $X$.


>>Then do we always have $\phi(t)$ is holomorphic with respect to $t$? If not, do we have counter examples?