Exact sequence for double mapping cylinder

Definition
Let $$X,Y,Z$$ be topological spaces and $$f:X \to Y, g: X \to Z$$ be continuous maps. Let $$D$$ be the double mapping cylinder of $$f$$ and $$g$$. Let $$i,j$$ denote the inclusions of $$Y$$ and $$Z$$ in $$D$$. Then we have the following long exact sequence of homology:

$$ \ldots \to H_q(X) \to H_q(Y) \oplus H_q(Z) \to H_q(D) \to H_{q-1}(X) \to \ldots$$

where the maps are:

$$a \in H_q(X) \mapsto (H_q(f)a, -H_q(g)a)$$

and:

$$(b,c) \in H_q(Y) \oplus H_q(Z) \mapsto H_q(i)b + H_q(j)c$$

And the third map is the usual connecting homomorphism from Mayer-Vietoris.

We can replace homology with reduced homology above.