If $A,B$ are homotopic equivalent, $C$ is any topological space then $[A,C]= [B,C]$

Question

Notation: $[A,C]= $ Set of homotopy classes of maps $A\to C$

Claim: If $A,B$ are homotopy equivalent spaces, $C$ is any topological space then $[A,C]$= $ [B,C]$

My attempt at a proof:

Suppose that $[h]\in [A,C]$ (where $[h]$ denotes a homotopy class of maps), so that $h:A\to C$ is a continous map.

$A,B$ being homotopy equivalent that means $\exists f:A\to B$ and $g:B\to A$ such that $f\circ g$ is homotopic to identity on $B$ and $g\circ f$ is homotopic to identity on $A$.

Then $h\circ g:B\to C$ and $(h\circ g)\circ f \simeq h$

I could not able to proceed further.

Any help would be appreciated.

Answer 1

Define $h\mapsto h\circ g$ in one direction and $h\mapsto h\circ f$ in the other : can you prove that they're inverse binections ?