# Are fully faithful functors left-cancellable up to natural isomorphism?

Let $F:C\to D$ be a fully faithful functor, so that for each pair of objects $c_1,c_2\in C$ we have a bijection:

$$F:\mathrm{Hom}_C(c_1,c_2)\to \mathrm{Hom}_D(F(c_1),F(c_2)).$$

Now let $G_1,G_2:B\to C$ be any functors and suppose that we have a natural isomorphism of functors $F\circ G_1\cong F\circ G_2$. I understand that this induces isomorphisms $G_1(b)\cong G_2(b)$ in $C$, one for each object $b\in B$, but are these isomorphisms natural in $b$? That is, do we have a natural isomorphism of functors $G_1\cong G_2$?

If not, what additional hypothesis on $F$ is necessary? Is there a standard name for this concept?

Sure. The natural isomorphism $\alpha_b: FG_1b\to FG_2b$ is the image of a unique $F^{-1}\alpha_b: G_1 b\to G_2b$ by full faithfulness. This is natural since $\alpha$ is, using faithfulness, and ann isomorphism since fully faithful functors are conservative.
A morphism $f$ in a 2-category with the property that isomorphisms $fg\cong fh$ are in natural bijection with isomorphisms $g\cong h$ is characterized by the property that the square whose upper legs are identities and whose lower legs are both $f$ is a so-called pseudo-pullback, so it would be reasonable to call $f$ a pseudo-mono. It's perhaps worth noting that fully faithful functors are also lax monos (remove the isomorphism condition in your statement,) and I think every lax mono is automatically pseudo.
• Well, this answer is about some kind of pseudo-monos. Strict monos, functors such that $FG=FH$ implies $G=H$, are just functors injective on objects and faithful. – Kevin Arlin Nov 14 '17 at 22:36