Definition 32.1 (Natural transformation):

Given a pair of functors $\pi:{\cal T}\longrightarrow {\cal G}$ and $H:{\cal T}\longrightarrow {\cal G}$, a natural transformation $T$ between $\pi$ and $H$ is a function which assigns to each object $X$ of ${\cal T}$ a morphism $\eta_X : \pi(X)\longrightarrow H(X)$ such that for each morphism $f : X \longrightarrow Y$ in ${\cal T}$, the following diagram commutes

$\begin{CD} \pi(X) @> {\pi(f)} >> \pi(Y) \\ @V{\eta_X}VV @VV{\eta_Y} V\\ H(X) @> H(f) >> H(Y) \\ \end{CD}$(32.12)

The notation used in this definition is quite suggestive. The Poincaré-Hurewicz map provides a natural transformation between the functors $\pi_1$ and $H_1$.