I'm currently reading Bott and Tu's "Differential Forms in Algebraic Topology". The authors introduce the Mayer-Vietoris Sequence as follows:
Suppose $M = U \cup V$, with $U,V$ open. Then there is a sequence of inclusions: $$ M \leftarrow U\coprod V \overset{\partial_0}{\underset{\partial_1}{\leftleftarrows}} U\cap V $$
Where $\partial_0,\partial_1$ are the inclusions of $U\cap V $ to $U$ and $V$ respectively. Applying the functor $\Omega^*$ to the sequence, we get a sequence of restricted forms $\dots$
I don't think an inclusion of the sort $ U \coprod V\hookrightarrow M$ is possible, and I'm not sure I understand what the authors meant. I think I understand what is the desired map in the differential forms level: $$ \omega \mapsto (i^{*}_U \omega , i^{*}_V \omega) $$
Is it possible to obtain the desired map by applying the functor $\Omega^*$ to a smooth map? if so, how?