Theorem 31.1:

(i) Suppose $f : X \longrightarrow Y$ and $g:Y\longrightarrow W$ are continuous functions,

$$H_n(g\circ f) = H_n(g)\circ H_n(f),\quad n = 0, 1, 2,\dots.$$

The identity map on $X$ induces the identity map on $H_n(X)$:

$$H_n($$id$$_X) =$$   id$$_{H_n(X)}$$

In other words the $\{H_n/ n = 0, 1, 2,\dots \}$ is a sequence of covariant functors from Top to AbGr. An immediate consequence is the following result.