Why are natural isomorphisms injective on objects?

Question

Here, here, and here says a natural isomorphism $\eta \colon F \rightarrow G$ can be regarded as a natural transformation with a two sided inverse, or alternatively each $\eta_X$ is an isomorphism. But what if $G$ isn’t one-to-one on the objects so two inverses are associated to some $G(X)$?

Answer 1

But what if $G$ isn’t one-to-one on the objects so two inverses are associated to some $G(X)$?

The inverse of $η_X \colon F(X) \to G(X)$ is not associated to $G(X)$, but to $X$.

Suppose that $F, G \colon \mathcal{C} \to \mathcal{D}$. The natural transformation $η$ consists of components $η_X \colon F(X) \to G(X)$, and these components are indexed by the objects $X$ of $\mathcal{C}$, not the objects $F(X)$ or $G(X)$ of $\mathcal{D}$. The natural transformation $η^{-1}$ consists similarly of components $(η^{-1})_X \colon G(X) \to F(X)$, and once again these components are indexed by the objects $X$ of $\mathcal{C}$, not the objects $G(X)$ or $F(X)$ of $\mathcal{D}$.