Theorem 29.3:

(i) For a continuous map $f : X \longrightarrow Y$, the maps $f_{\sharp}:S_n(X)\longrightarrow S_n(Y)$ satisfy

$$\partial _n\circ f_{\sharp} = f_{\sharp}\circ \partial _n \eqno(29.7)$$

The $f_{\sharp}$ on the right hand side obviously refers to the map $S_{n-1}(X)\longrightarrow S_{n-1}(Y)$ and $\partial _n$ refers to the boundary operator on $S_n(Y)$ on the left hand side whereas it refers to the boundary operator on $S_n(X)$ on the right hand side.

(ii) If $f : X \longrightarrow Y$ and $g : Y \longrightarrow Z$ are two continuous maps then the maps $f_{\sharp}:S_n(X)\longrightarrow S_n(Y)$ and $g_{\sharp}:S_n(Y)\longrightarrow S_n(Z)$ satisfy

$$(g\circ f)_{\sharp} = f_{\sharp}\circ g_{\sharp}\eqno(29.8)$$