Theorem 34.8 (Naturality of the Mayer Vietoris sequence):

Given a continuous map of pairs $f:(U, V)\longrightarrow (A, B)$ where $\{U, V\}$ and $\{A, B\}$ are open coverings of topological spaces, the following diagram commutes where the vertical maps are induced by $f$.

$\begin{CD} @> >> H_n(U\cap V) @> >> H_n(U)\oplus H_n(V) @> >> H_n(U\cup V) @>... ...H_n(A)\oplus H_n(B) @> >> H_n(A\cup B))@> >> H_{n-1}(A\cap B)@> >> \\ \end{CD}$(34.13)