I am reading Stong's notes on Cobordism, and he defines a Cobordism category $\mathcal{C}$, with direct sums and a boundary functor $\partial:\mathcal{C}\longrightarrow \mathcal{C}$, and says that two objects $X,Y\in\mathcal{C}$ are cobordant if there exist a pair of objects $U,V\in \mathcal{C}$, such that $X+\partial U \cong Y + \partial V$ in $\mathcal{C}$.
The classical definition of cobordism says that two manifolds with boundary $X,Y$ are cobordant if there exists a manifold $M$ whose boundary $\partial M$ is isomorphic to the disjoint union of $X$ and $Y$.
Stong says that the two definitions are equivalent. In the one direction, $X\sqcup Y = \partial M$ implies that $X + \partial M\cong Y + \partial (X\times I)$, where $I = [0,1]$. This trivially follows from the definition of $\partial (X\times I)$.
On the other direction, he says that if there is a $U$ and $V$ such that $X + \partial U\cong Y + \partial V$ then there exists an $M$ such that $\partial M = X\sqcup Y$ due to an easy geometric construction, but I can't see it.
One thing is that the morphisms in the cobordism category send boundaries to boundaries, so I imagine that if there is an isomorphism between $X + \partial U$ to $Y + \partial V$, then this should somehow map $\partial U$ to $\partial V$ isomorphically, but a priori it is unclear that $\partial U$ cannot be mapped in part to $\partial Y$.
Any help with proving this direction would be appreciated.
EDIT: Stong defines "being cobordant" as an equivalence relation in the Cobordism category, and says that closed objects ($X\in \mathcal{C}$ with $\partial X\cong \emptyset$) under this relation form something he calls the cobordism semigroup. The statement is then that two objects in the cobordism semigroup are cobordant according to his definition if and only if they are cobordant w.r.t. Thom's definition. The proof of the second direction appears in another stackexchange question for oriented cobordism, but it is just the same for unoriented cobordism.