I'm reading Gauduchon's paper Hermitian connections and Dirac operators.

For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = \nabla J=0$.

The torsion $T$ of $\nabla$ is seen in the space $\Omega^2(T M)$ of vector-valued 2-forms which in turn is seen as a space of trilinear forms that are skew wrt the last two arguments by the identification $T(X,Y,Z) = g(X, T(Y, Z))$.

At the beginning of section 2 the paper says $A(g, J)$ is an affine space modeled on the space $\Omega^{1,1}(T M) = \{ B \in \Omega^2(T M): B(JX, JY) = B(X,Y) \}$.

For a fixed $\nabla \in \mathcal A(g, J)$ define its potential wrt the Levi Civita as $\nabla_X Y - D_X Y =: A^\nabla(X,Y)$.

Is it that obvious that, given $\nabla_1, \nabla_2 \in \mathcal A(g, J)$ then the difference satisfies $$ A^{\nabla_1}(JX, JY) - A^{\nabla_2}(JX, JY) = A^{\nabla_1}(X,Y) - A^{\nabla_2}(X,Y) ?? $$

Thanks

David