In this talk, we use the canonical connection instead of Levi-Civita connection to study the smooth maps between almost Hermitian manifolds, especially, the pseudoholomorphic ones. By using the Bochner formulas, we obtain the $C^2$-estimate, Liouville type theorems of pseudoholomorphic maps, pseudoholomorphicity of pluriharmonic maps, Simons integral inequality and bounds of half norm of parallel canonical second fundamental form of pseudoholomorphic isometry.