"Standard arguments" in Mahowald's eta_j paper

The short answer is that composition in Ext does correspond to composition of the maps, if nothing intervenes. In the case in hand, if $p$ is the projection of $X_j$ onto its top cell, then $f_j$ is represented by an element $a \in Ext^1$ such that $p_*(a) = h_j$, and $g_j$ is represented by $p^*(h_1)$, so that $f_jg_j$ is represented by $ap^*(h_1) = p_*(a)h_1 = h_jh_1$.

For more detail about Mahowald's maps, and some simplification in their construction, see my paper with David Hunter: [David Hunter and Nicholas Kuhn, Mahowaldean families of elements in stable homotopy groups revisted, Math. Proc. Camb. Phil Soc. 127 (1999), 237-251]. I think we did a good job in breaking the construction down into understandable bits. (Unfortunately our paper is just too old to be on the Arxiv.)

Since the paper actually refers to a secondary operation associated with the Adem relation $$Sq^{2^i+1}Sq^1+Sq^2Sq^{2^i}+Sq^4Sq^{2^i-2}+Sq^{2^i}Sq^2=0$$ then a standard argument to show that the composition $S^{2^i}\stackrel{g_i}\to X_i\stackrel{f_i}\to S^0$ is essential is to proceed and prove by contradiction. If the composition is null then one may look at the cohomology of the double mapping cone $S^0\cup_{\overline{f_i}}C(X_i\cup e^{2^i+1})$ where existence of a map $\overline{f_i}:C_{g_i}\to S^0$ follows from the assumption that $f_i\circ g_i$ is null. Now, evaluation of the above Adem relation on the $0$-dimensional class of in the cohomology of the double mapping cone gives the desired contradiction. This is a standard argument prior to the use of Adams Spectral Sequence arguments.