We can, of course, define the functor category $[F,G]$ by considering the natural transformations between two functors $F$ and $G$.
But, suppose that instead of the natural transformations we consider the dinatural transformations between two (appropriately defined) functors. Can we define a category structure on this set? More specifically, do we have a "nice" way to compose two dinatural transformations? Thanks!
The answer is not, to explain why let me recall briefly the definition of dinatural transformation.
Given two parallel functors $F,G \colon \mathbf C^\text{op} \times \mathbf C \to \mathbf X$ a dinatural transformation $\tau \colon F \to G$ is a family of morphisms $\tau_c \colon F(c,c) \to G(c,c)$ such that
$$G(c,f)\circ \tau_c \circ F(f,c) = G(f,c') \circ \tau_{c'} \circ F(c',f)$$
which can of course be described in term of commutative diagrams.
Now given $F,G,H \colon \mathbf C^\text{op} \times \mathbf C \to \mathbf X$ and $\tau \colon F \to G$ and $\sigma \colon G \to H$, while is true that is possibile to obtain a family $\sigma_c \circ \tau_c \colon F(c,c) \to H(c,c)$ it isn't true that this is a dinatural transformation, at least not in general (the commutative diagrams of dinaturality cannot ensure di naturality of the composed morphisms $\sigma_c \circ \tau_c$).
That's the main problem to make dinatural transformation in morphisms of a category.