My professor in his notes claims that a functor $G$ has a right adjoint iff the functor $Y \mapsto Hom(X, G(Y)$ is corepresentable, i.e for each $X$ there is an object $F(X)$ and a nutural by $X$ bijection
$Hom(F(X), Y) \cong Hom(X, G(Y))$.
Is that true? Of course, from corepresentability follows that $X \mapsto F(X)$ is indeed a functor (functoriality by $X$ plus Yoneda lemma gives us a unique arrow $F \phi : F(X) \to F(X')$ for each $\phi: X \to X'$).But I don't see how do we get functoriality by Y.
If it was true it would be enough to check if two given functors is adjoint pair to verify functoriality just by one argument. I never heard of it and always checked both arguments.