(natural transformation)
Let \(F, G: \mcc \to \mcd\) be covariant functors. A natural transformation \(F \to G\) is the data of maps \(F(X) \to G(X)\) in \(\mcd\) for all \(X \in \mcc\), such that for every \(f: X \to Y\) in \(\mcc\) the square \[ \begin{matrix} F(X) & \to & G(X) \\ \downarrow & & \downarrow \\ F(Y) & \to & G(Y) \end{matrix} \] commutes. For contravariant functors one asks the analogous square, with the vertical arrows reversed, to commute.
(natural isomorphism)
A natural isomorphism \(F \to G\) is a natural transformation such that \(F(X) \to G(X)\) is an isomorphism for every object \(X\).
Natural transformations are the morphisms of the functor category \(\mcd^{\mcc}\), and the natural isomorphisms are its isomorphisms: if each component \(F(X)\to G(X)\) is an isomorphism, the inverse maps \(G(X) \to F(X)\) form a natural transformation.
On finite-dimensional vector spaces over a field \(k\), the evaluation maps \(V\to V^{\vee\vee}\) form a natural isomorphism from the identity functor to the double-dual functor. An isomorphism \(V\cong V^\vee\) exists for each \(V\) but depends on a choice, such as a basis; the dual is contravariant, so the identity functor and \(V\mapsto V^\vee\) are not functors of the same variance.
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