Let $\cal F$ be a functor from the category of topological spaces to some category $\cal C$ Then $\cal F$ is called homotopy invariant if for any two homotopic maps $f,g\colon X\to Y$ between topological spaces $X$ and $Y$ the morphisms ${\cal F}f$ and ${\cal F}g$ in $\cal C$ induced by $\cal F$ are identical.

Suppose $\cal F$ is a homotopy invariant functor, and $X$ and $Y$ are homotopy equivalent topological spaces. Then there are continuous maps $f\colon X\to Y$ and $g\colon Y\to X$ such that $g\circ f\simeq{\rm id}_X$ and $f\circ g\simeq{\rm id}_Y$ (i.e. $g\circ f$ and $f\circ g$ are homotopic to the identity maps on $X$ and $Y$ respectively). Assume that $\cal F$ is a covariant functor. Then the homotopy invariance of $\cal F$ implies $$ {\cal F}g\circ{\cal F}f={\cal F}(g\circ f)={\rm id}_{{\cal F}X} $$ and $$ {\cal F}f\circ{\cal F}g={\cal F}(f\circ g)={\rm id}_{{\cal F}Y}. $$ From this we see that ${\cal F}X$ and ${\cal F}Y$ are isomorphic in $\cal C$ (The same argument clearly holds if $\cal F$ is contravariant instead of covariant.)

An important example of a homotopy invariant functor is the fundamental group $\pi_1$ here $\cal C$ is the category of groups.