Terminology question: what does a natural isomorphism do to maps?

Question

Suppose I have categories $C$ and $D$ and naturally isomorphic functors $F,G\colon C \to D$. (I do. Trust me.) Now name the natural isomorphism $\theta$; then for any arrow $f\colon x \to y$ in $C$, there are isomorphisms $\theta_x\colon F(x) \to G(x)$ and $\theta_y\colon F(y) \to G(y)$ such that $$F(f) = \theta_y^{-1} \circ G(f) \circ \theta_x.$$

Morally, $\theta$ "takes $F(f)$ to $G(f)$."

I want to be able to say "$F(f)$ is naturally isomorphic to $G(f)$." But I think that phrasing is not kosher. The briefest coherent way I think to phrase this relationship is that "$F(f)$ and $G(f)$ are conjugate through natural isomorphisms." But that sentence is a stylistic monstrosity. So ...

What does one say?

Answer 1

I would say that the maps are naturally equivalent. You're right that saying naturally isomorphic is kinda abusing the terminology but it doesn't sound monstrous to me. Whatever you choose you should just make sure to explain it the first time you use it.

Answer 2

It is no abuse of notation to say that $F(f)$ and $G(f)$ are isomorphic:

$F(f)$ is isomorphic to $G(f)$ in the arrow category of $D$.