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> Let $F: \mathcal C \to \mathcal D$ and $G: \mathcal D \to \mathcal C$ be quasi-inverse functors, and let $H : \mathcal C \to Set$ be a representable (contravariant) functor with representative $X \in \mathcal C$. Prove that $H \circ G$ is representable by $F(X)$.
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$\DeclareMathOperator\Hom{Hom}$As ismorphisms are transitive, it suffices to consider the case when $H = \Hom( -, X)$. To this end, we wish to find $\phi : \Hom(-,X) \circ G \to \Hom(-,F(X))$ an ismorphism, from which we quickly deduce that for any $f: B \to A$ and $g: GA \to X$ it must be that $\phi\_A(g) \circ f = \phi\_B(g \circ Gf)$. I am not sure how to dind such a $\phi$ though. It seems like I have to somehow use the fact that $F$ and $G$ are quasi-inverses...