What does it mean when a diagram "commutes to homotopy"?

Question

What does commutes in $hTop_*$ mean and how does the proof following this sentence show it?

The proof says this follows from $q_1m \simeq 1_x \simeq q_2m$ where $q_1, q_2 : X \lor X \rightarrow X$ are defined as projections that send $x$ to $(x,x_0)$ or $(x_0,x)$, respectively and $X \lor X$ is viewed as the subspace of $X \times \{x_0\} \cup \{x_0\} \times X$.

Answer 1

It means that the functions $\Delta$ and $k \circ m$ are (base-point preserving) homotopic.

Answer 2

I'm used to seeing it as (and would write it as) "commutes up to a homotopy". The phrase 'up to an X' means we are setting a weaker standard for equality than strict equality. I've also seen it used in the phrase "$\mathbb{Z}_2 \times \mathbb{Z}_3$ is the same as $\mathbb{Z}_6$ (up to a group isomorphism)".