What do you call the relationship between two functions that perform essentially the same operation over isomorphic objects?

Question

This question applies to any category, but I'm going to use vector spaces as an easy example. Suppose we let $V$ be the space of column vectors in $\mathbb R^3$, and consider the dual space $V^*$ as the space of row vectors in $\mathbb R^3$. Define two functions $f \colon V \to V$ and $g\colon V^* \to V^*$ given by $$\begin{pmatrix} a \\ b \\ c \end{pmatrix} \overset{f}{\mapsto} \begin{pmatrix} 2a \\ 2b \\ 2c \end{pmatrix}$$ and $$\begin{pmatrix} a & b & c \end{pmatrix} \overset{g}{\mapsto} \begin{pmatrix} 2a & 2b & 2c \end{pmatrix}.$$ Obviously we can't say that $f$ and $g$ are the same function because they operate over different sets, but it is still clear that they "perform the same operation" over their respective sets. Put more rigorously, we can say that $V \cong V^*$ under the isomorphism $\varphi\colon V \to V^*$ given by $$\begin{pmatrix} a \\ b \\ c \end{pmatrix} \overset{\varphi}{\mapsto} \begin{pmatrix} a & b & c \end{pmatrix},$$ and then we can say that $g(\varphi(v)) = \varphi(f(v))$ and $f(\varphi^{-1}(w)) = \varphi^{-1}(g(w))$ for all $v \in V$ and $w \in V^*$. Is there a general term used to describe this relationship between $f$ and $g$?

Answer 1

Here's a variety of possibilities, though none of them could be used without comment.

First, $f$ and $g$ are in fact isomorphic themselves when viewed as objects in an arrow category. However, without restriction this is probably a more inclusive notion than you want.

Next, the transformation that takes $f$ to $g$ via the isomorphism $V\cong V^*$is called a conjugation action. Similarity in linear algebra is a special case of this. You could, by analogy, say that $g$ is a conjugate of $f$, though that would be ambiguous in many contexts and certainly would need some explanation.

In the particular setting you use, as mentioned in the previous paragraph, (the matrices representing) $f$ and $g$ are similar.

Ultimately, I personally would just spell out what I meant e.g. "$g = \varphi\circ f\circ\varphi^{-1}$ for some isomorphism $\varphi$".