# Parity reversed tangent bundle of a supermanifold and the corresponding Q-structure

I asked this question in MathStackExchange 10 days ago but get no response (not a vote nor a comment), so I'm copying it here below. The link to the original question is:

https://math.stackexchange.com/questions/4507710/parity-reversed-tangent-bundle-of-a-supermanifold-and-the-corresponding-q-struct

Given an ordinary manifold $$X$$, the pair $$(X,\Omega^\bullet)=:\Pi TX$$ is called the parity reversed tangent bundle of $$X$$, which is a supermanifold. The page 4 of AKSZ's The Geometry of the Master Equation and Topological Quantum Field Theory gives, as an example of $$Q$$-manifold, the parity reversed tangent bundle $$\Pi TN$$ ($$N$$ an ordinary manifold) equipped with the odd vector field $$Q$$ defined in local coordinates $$Q = \eta^a\frac{\partial}{\partial x^a},$$ where $$x^a$$'s are the coordinates on $$N$$ and $$\eta^a$$'s are the corresponding coordinates along the fibre. After this it writes that "Note that $$\Pi TN$$ can be defined and equipped with a $$Q$$-structure also in the case where $$N$$ is a supermanifold" without any further explanation.

The question is that I'm not sure what is the definition of $$\Pi TN$$ when $$N$$ is a supermanifold. I did come up with one definition myself, but it seems too trivial to be the proper one. The following is the definition that I've came up with:

'Let $$\mathcal{M}=(M,C^\infty_\mathcal{M})$$ be a supermanifold and $$C_M^\infty$$ (not $$C^\infty_\mathcal{M}$$) denote the ring of ordinary smooth functions on $$M$$. The parity reversed tangent bundle of $$\mathcal{M}$$ is the supermanifold $$\Pi T\mathcal{M}= (M,C^\infty_\mathcal{M}\otimes_{C^\infty_M} \Omega^\bullet),$$ where $$C^\infty_\mathcal{M}\otimes_{C^\infty_M} \Omega^\bullet$$ is the tensor product of $$C^\infty_\mathcal{M}$$ and $$\Omega^\bullet$$ seen as sheaves on $$M$$. '

With this definition, the mentioned $$Q$$-structure is obvious, with the defining formula exactly the same as above. But this doesn't look correct to me, since the supermanifold has a super tangent bundle on its own right, while this definition uses only the tangent bundle of the base manifold (with the induced smooth structure): the derivations along odd coordinates are nowhere included, and I don't know how to fix this.

I'd appreciate it if anyone could tell me the true definition, or convince me that the definition I came up by myself is the correct one (if it really is). If the latter is not the case, it would be even better if hints about the $$Q$$-structure could be given, since the structure might be no longer obvious under the new definition.