Homotopic to wich identity function

Question

I'm having some problem understanding when a function is homotopic to the identity function.
Let's say that i have the function $\Delta:X\to X\times X$ such as $\Delta(x)=(x,x)$ and I want to show that it's homotopic to the identity function, but i can't understand which identity function I'm talking about.
From the definition I have two function $f,g:X\to Y$ are homotopic to each other if $\exists H:X\times[0,1]\to Y$ such as $H(x,0)=f(x)$ and $H(x,1)=g(x)$, so $\Delta$ can't be homotopic to $id_X$ but neither to $id_Y$.
But at the same times I can find two function $\Delta:X\to X\times X$ and $i:X\times X\to X$ to show that $X\simeq X\times X$.
Am I making any mistakes? Is there a way to show that $\Delta$ is homotopic to one of those identity function?
Can I show that $id_X\simeq id_{X\times X}$?

Answer 1

No, a continuous map $X\to Y$ can't be homotopic to "the identity" when $X\neq Y$. Maps $X\to Y$ can only be homotopic to other maps $X\to Y$. However, when you have maps $f\colon X\to Y$ and $g\colon Y\to X$ you obtain $f\circ g\colon Y\to Y$ and $g\circ f\colon X\to X$ which can be homotopic to $\operatorname{id}_Y$ and $\operatorname{id}_X$, respectively. In this case $f$ (and $g$) is called a homotopy equivalence between $X$ and $Y$ and we write $X\simeq Y$.