Suppose $ f : X \to Y$ and $g,h : Y \to Z$ are continuous, with $g \simeq h$. Prove that $ gf \simeq hf $.
My attempt:
Suppose $ L(x,t) $ gives a homotopy from $g$ to $h$, i.e. $ L(x,t) : Y \times I \to Z $, with $ L(x,0) = g(x)$ and $ L(x,1) = h(x) \ \forall x \in X$.
Then let $ H(x,t) : X \times I \to Z $ be defined by $ H(x,t) = L(f(x),t) $. Then $H$ is continuous, since it is the composition of continuous maps. $H(x,0) = L(f(x),0) = g(f(x)) $ and $ H(x,1) = L(f(x),1) = h(f(x)) $. This $H$ is the desired homotopy.
Is this correct?
Thanks!