Let $X$ and $Y$ be complex manifolds with canonical almost complex structures $J_X: TX \to TX$ and $J_Y: TY \to TY$. Let $F: X \to Y$ be a smooth map. Then $F$ is holomorphic if and only if, for every point $p \in X$, the tangent map $dF_p: T_pX \to T_{F(p)}Y$ is complex linear with respect to the almost complex structures, equivalently

\begin{align*} dF_p \circ J_{X,p} = J_{Y,F(p)} \circ dF_p. \end{align*}