In the book introduction to topological manifolds by John M. Lee, we have the following proposition (p. 345):
Let $X$ be a topological space and $\{X_\alpha\}_{\alpha \in A}$ the set of path components of $X$. Then $$H_n(X) \cong \bigoplus_{\alpha \in A}H_n(X_\alpha)$$ holds for all $n \in \mathbb{N}$.
Now the proof starts as follows: for all $\alpha \in A$, the inclusions $\iota_\alpha : X_\alpha \to X$ induce isomorphisms $$\bigoplus_{\alpha \in A} C_n(X_\alpha) \to C_n(X)$$ My question is, how exactly do they induce isomorphisms, or more generally a homomorphism?
Well, each $\iota_\alpha$ induces a homomorphism $C_n(X_\alpha)\to C_n(X)$. Combining these homomorphisms together, we get a homomorphism $\bigoplus_\alpha C_n(X_\alpha)\to C_n(X)$. Explicitly, an element of $\bigoplus_\alpha C_n(X_\alpha)$ has finitely many nonzero coordinates, each of which is a formal linear combination of maps $\Delta^n\to X_\alpha$ for different values of $\alpha$. Given such an element, we compose all of these maps with the inclusion maps $X_\alpha\to X$ to get maps $\Delta^n\to X$. So we have finitely many formal linear combinations of maps $\Delta^n\to X$, which we can just add together to get a single formal linear combination of maps $\Delta^n\to X$, i.e. an element of $C_n(X)$.
To see that this is an isomorphism, note that every map $\Delta^n\to X$ factors through $X_\alpha$ for exactly one $\alpha$ (since its image must be contained in a path-component of $X$). So every element of $C_n(X)$ can uniquely be written as a sum of elements which come from $C_n(X_\alpha)$ for different values of $\alpha$: given a chain, just split it up into its simplices which are contained in the different path-components of $X$.