A $(n+1)$-topological quantum field theory $\mathcal{T}$ is a rigid symmetric monoidal functor from the category **$(n+1)$-Cob** of $n$-manifolds and $(n+1)$-cobordisms to **FdVect**.
My question is about the symmetric monoidal structure of **$(n+1)$-Cob** and how it plays well with functoriality:

We consider our manifolds abstractly (as opposed to being embedded in some ambient space). The monoidal structure in **$(n+1)$-Cob** is given by disjoint union, thus
$$M\otimes N = M \sqcup N$$
for manifolds $M,N$, and likewise for cobordisms. But since our manifolds are just abstract, surely this is exactly the same object as
$$N\otimes M=N\sqcup M.$$
But when we pass to **FdVect** our functor gives
$$ \mathcal{T}(M\otimes N)=\mathcal{T}M \otimes \mathcal{T}N $$
while
$$ \mathcal{T}(N\otimes M) = \mathcal{T}N \otimes \mathcal{T}M $$
which are certainly isomorphic but distinct objects.

So clearly something is broken in my understanding of this, but what?
Do we somehow distinguish between $M\otimes N$ and $N\otimes M$ in **$(n+1)$-Cob**?
If so, how, given that these are just abstract manifolds?

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