What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the bijections are natural in $x$ and not in $y$?

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If there are bijections natural in $x$, then $F$ must have a right adjoint, whose object part agrees with $G$ and whose morphism part makes the bijection natural in $y$. –  Zhen Lin Feb 11 '14 at 21:40
@ZhenLin, you have a reference for that fact? –  Mariano Suárez-Alvarez Feb 11 '14 at 21:51
See Corollary 2 in [CWM, Ch. IV, §1]. –  Zhen Lin Feb 11 '14 at 22:38

Let $C$ and $D$ both be the category of finite-dimensional (say real) vector spaces and invertible linear maps between them, let $F$ be the identity, and let $G$ take a vector space to its dual. ($G$ is not functorial on all linear maps, but it is on the invertible ones.) Then $G y \cong y$ unnaturally, so $\hom(F x, y) = \hom(x, y) \cong \hom(x, G y)$ naturally in $x$ but unnaturally in $y$.
To connect this answer with Zhen Lin's comment on the question: The corollary that Zhen Lin cited requires the axiom of choice (unless we allow $G$ to become an anafunctor), so it's unsurprising that I rely on it: choose for each $y$ a linear isomorphism $d_y\colon y \to G y$ (or choose for each $y$ a basis if you want to feel more elementary). Then given a linear isomorphism $f\colon y \to y'$, let $G f$ be $d_y \circ f \circ d_y^{-1}$. Now $\hom(F x, y) \cong \hom (x, G y)$ naturally in $y$ too. (My original $G f$ is meant to be the dual of $f^{-1}$.) –  Toby Bartels Feb 12 '14 at 17:59