Given two functions $f,g$, I need to find a function $h$ such that $h\circ f=g\circ h$.
For example, if $f(x)=x+1$ and $g(x)=x-1$ where $f,g:\Bbb Z\to\Bbb Z$. Then $h$ must have the property that $h(f(x))=g(h(x))\iff h(x+1)=h(x)-1$. Here it is easy to see that if $h(x)=-x$ the property holds, because $-(x+1)=-x-1$.
Another example is if $f(x)=x+1$ and $g(x)=2x$, then $h(x+1)=2h(x)$ must be true, and if $h(x)=2^x$, then $h$ has the said property.
But in this examples $h$ is found just intuitively, is there a general way to find $h$ for any given two functions $f$ and $g$? If there is not a general way, is there some specific way with specific choices of $f$ and $g$? Thanks.