If $f$ and $g$ are homotopic maps from $X$ to $Y$ and $p : W \rightarrow X$ and $q: Y \rightarrow Z$ are any maps, then $f \circ p$ is homotopic to $g \circ p$ and $q \circ f$ is homotopic to $q \circ g$.
I see that if $H$ is homotopy from $f$ to $g$, then $H'(x,t) = q \circ H(x,t)$ is the required composition of functions since $H:X \times I\rightarrow Y$ and $q:Y \rightarrow Z$.
But I can't figure out how to find a map for $f \circ p$ and $g \circ p$. It looks likes $H(x,t) \circ p$ would be a good map, but this is impossible because $p : W \rightarrow X$ and $H:X \times I \rightarrow Y$ where the domains are not the same set. The book I'm using says $H \circ (p \times I_{d_{I}})$ is a desired map, but I am completely unaware of what $p \times I_{d_{I}}$ means for maps.
Anyone have any ideas about a map or what these symbols mean?
The maps $p\colon W\rightarrow X$ and $q\colon Y\rightarrow Z$ are at least required to be continuous.
The map $F:=H\circ(p\times\textrm{id})$ is defined by: $$F(x,y)=H(p(x),y).$$ A proper notation could have been $H\circ(p,\textrm{id})$, but it is pretty common to write $\times$ or $\otimes$ for this operation.