Natural Transformation between covariant and contravariant functor Suppose $F: C \rightarrow D$ is a covariant functor and 
$G: C \rightarrow D$ is contravariant. Does it make sense to define
natural transformations $\eta_1 : F \Rightarrow G$ or 
$\eta_2 : G \Rightarrow F$ or are there any obstructions? 

It does make sense. If $F$ is covariant and $G$ is contravariant, a transformation $F \to G$ is a family of maps $F(x) \to G(x)$ such that for every morphism $x \to y$ the diagram
$$\begin{array}{ccc} F(x) & \rightarrow  & G(x) \\ \downarrow && \uparrow \\ F(y) & \rightarrow & G(y) \end{array}$$
commutes. In fact, this is a special case of the more general notion of dinatural transformations.
