Suppose we have a smooth function $f:\mathcal{M}\times\mathcal{N}\rightarrow\mathbb{R}$ where the domain is a product of two Riemannian manifolds. The Riemannian cross-derivative ([1], section 2) is defined as: $$ \mathrm{grad}_{xy}^2:= \mathrm{D}_x\mathrm{grad}_y f(x,y): \mathrm{T}_x\mathcal{M}\rightarrow \mathrm{T}_y\mathcal{N} $$

I'm trying to understand this operator. I have two specific questions:

First, to me, if we fix $y$ then $\mathrm{grad}_y f(x,y):\mathcal{M}\rightarrow\mathrm{T}\mathcal{N}$, and the differential on $x$ would result in an operator $\mathrm{D}_x\mathrm{grad}_y f(x,y):\mathrm{T}\mathcal{M}\rightarrow\mathrm{T}\mathrm{T}\mathcal{N}$, which seems not the same as the definition in the paper. Is there any point I missed here?

Second, I'm trying to understand it similar to its Euclidean counterparts. Under what curcumstances do we know that $\mathrm{grad}_{xy}^2$ and $\mathrm{grad}_{yx}^2$ are adjoints? That's to say, do we have: $$ \langle \eta,\mathrm{grad}_{xy}^2(\xi) \rangle_{y} = \langle\mathrm{grad}_{yx}^2(\eta),\xi \rangle_{x}, \forall \xi\in \mathrm{T}_{x}\mathcal{M}\text{ and }\forall \eta\in \mathrm{T}_{y}\mathcal{N} $$

Any comments or references are appreciated, thanks!

References:

[1] Han, Andi, et al. "Riemannian Hamiltonian methods for min-max optimization on manifolds." SIAM Journal on Optimization 33.3 (2023): 1797-1827.