Let $M$ be a compact manifold with dimension $\geq 3$. By a theorem of ELIASHBERG,

the cotangent bundle of $M$ admits an integrable complex structure $J$ such that

$(T^*M, J)$ is a stein manifold. My question is: Define the map $f$ by

$$f: T^*M \rightarrow T^*M, (x,v) \rightarrow (x, -v), x \in M, v \in T_x^*M$$. Is it true that $f$ is an anti-holomorphic map with respect to the complex structure $J$ on $T^*M$?