Classifying maps are natural up to based homotopy

Take CW structures with the basepoints as vertices and choose a maximal tree $T$ in the connected one-skeleton of $BG$, using [A1]. Send all vertices and $T$ to the basepoint of $BH$. For each oriented edge $e$ outside $T$, the tree paths to its endpoints turn $e$ into a based loop; send $e$ to a loop representing its image under $\varphi$. These loops generate ${\pi }_1(BG)$, and each two-cell attaching loop maps to the identity because $\varphi$ is a homomorphism on $G$. Thus the map extends over the two-cells. For a cell of dimension $k\ge 3$, its attaching sphere extends because ${\pi }_{k-1}(BH)=0$ by [F1]. AC chooses the tree, loop representatives, and cell extensions. The weak topology of the CW structure makes these compatible cellwise maps continuous, giving the required based map $B\varphi$.

Write $a=B\varphi \circ c_X$ and $b=c_Y\circ f$, so $a_{\ast }=b_{\ast }=\varphi$. Choose a maximal tree in $X^{(1)}$ rooted at $x_0$, and let ${\alpha }_v$ be its path from $x_0$ to each vertex $v$. Choose the vertex track from $a(v)$ to $b(v)$ to be $a({\alpha }_v{)}^{-1}b({\alpha }_v)$, with the track at $x_0$ constant. For an edge $e:v\to w$, the boundary of its homotopy square, conjugated by $a({\alpha }_v)$, represents $a_{\ast }[{\alpha }_{v}e{\alpha }_w^{-1}]b_{\ast }[{\alpha }_{v}e{\alpha }_w^{-1}{]}^{-1}=1$. Hence every such square fills. For each $k$-cell with $k\ge 2$, the boundary of its homotopy prism is a sphere $S^k\to BH$, which fills because ${\pi }_k(BH)=0$ by [F1]. AC chooses the fillings and the CW weak topology assembles them into a continuous homotopy fixed at $x_0$. Thus $a\simeq b$ by a based homotopy.

By [F3], the homotopy of step 2.1 gives $(B\varphi {)}_{\ast }\circ (c_X{)}_{\ast }=(c_Y{)}_{\ast }\circ f_{\ast }$. This is the asserted naturality, including degree two.