Given a map $f:A\to B$, define the map $p_n(f):A^n\to B^n$ as $p_n(f)(a_1,...,a_i)=(f(a_1),...,f(a_i))$. Equivalently, you could say given $f$, $p_n(f)$ is such that $\mathrm{proj}_a\circ p_n(f)=f\circ \mathrm{proj}_a$.
My question is: is there a notation for $p_n(f)$ where we don't have to go through the the hassle of explaining what it is? I thought maybe some kind of tensor product or direct sum of $f$ would do the trick but I don't think they define it in the way I need.
I guess you could write $p_n(f)=(f(\cdot),...,f(\cdot))$ but I don't find that very asthetically pleasing. Is there another notation that could do this?
Note: if it helps, I'm working with the specific case where $f$ is invertible.