Exercise 1.9 on page 31 of Kashiwara's Categories and Sheaves asks:
Let $F: C \to C'$ be an equivalence of categories and let $G$ be a quasi-inverse. Let $H: C \to \text{Set}$ be a representable functor, $X$ a representative. Prove that $H\circ G$ is representable by $F(X)$.
I think I solved it as follows: suppose $\omega \in \text{Hom}_{[C,\text{Sets}]}(H, \text{Hom}(X, \cdotp))$ is an isomorphism, and suppose $\theta: \text{Hom}_{C}(\cdotp, G(\cdotp)) \xrightarrow{\sim} \text{Hom}_{C'}(F(\cdotp), \cdotp)$ is the adjunction. Then $\theta_{X, -}^{-1}\overset{\text{V}}{\circ}(\omega \overset{\text{H}}{\circ} \text{Id}_G)$ is an isomorphism from $H \circ G$ to $\text{Hom}_{C'}(FX, \cdotp)$, where I've written V for vertical composition and H for horizontal composition.
My question is: why did they assume $F$ was an equivalence? It didn't seem to enter into the proof at all. It seems sufficient to assume that $F$ has a right adjoint.
Sorry to ask a silly question; I just want to make sure I'm getting this stuff.
you are correct, it's enough that $F$ has a right adjoint $G$, since the composition of natural isomorphisms gives you the representation, given $Y\in C'$ then $HG(Y)=[X,GY]=[FX,Y]$