Does there exist a definition of equivalence of functors? I have two functors $F_1,F_2$ from a category $C$ into two distinct categories $D_1,D_2$. I would like to say that $F_1$ and $F_2$ are equivalent if there exists a commutative square
$\require{AMScd}$
\begin{CD}
C @>F_1>> D_1\\
@V F_2 V V @VV E_1 V\\
D_2 @>>E_2> D
\end{CD}
of functors such that the $E_i$ are equivalences. Can one say that this notion is well-known?:) Do there exist any similar notions in the literature?
Upd. Possibly, it would be better to reverse the arrows here, that is, to demand that there exists a functor $F$ (that is bijective on objects) and two equivalences of categories $E_1$ and $E_2$ such that $F_i=E_i\circ F$.
 A: It's worth first understanding the 1-categorical analogue: what would it mean for two arrows $f \colon X \to Y$ and $f' \colon X \to Y'$ in a category $C$, which share the same domain, to be isomorphic? One way of making sense of this is by saying that they are isomorphic in the coslice category $X/C$, whose objects are pairs $(Y,f)$ where $Y$ is an object of $C$ and $f \colon X \to Y$ is an arrow; morphisms are commutative triangles.
Your question is really about the analogue of the above when $C$ is not a 1-category but the 2-category $Cat$. Now there are multiple inequivalent ways of defining a "coslice 2-category" under an object of a 2-category, depending on whether you want the 2-morphisms to be invertible, or identities, or if they are not invertible, in which case you need to choose a direction for them. You want invertible 2-cells, which is also what people will generally expect by default if you only say "coslice 2-category". More precisely, the following are equivalent:

*

*there exists a square as in your question which commutes strictly

*there exists a square as in your question which commutes up to a natural isomorphism

*there exists a commuting triangle which commutes up to a natural isomorphism

However, asking for a triangle which commutes strictly is in general strictly stronger (and seems like a rather unnatural notion).
https://ncatlab.org/nlab/show/slice+2-category
