Identity as a natural transformation

A rather trivial but nice result: If $𝖢$ is a category, $1 𝖢 : 𝖢 \to 𝖢$ is the identity functor and $i d \in 𝖢 𝖢 (1 𝖢, 1 𝖢)$ is taken to donate the corresponding Identity natural transformation, then the notation $i d 𝑋$ for the identity morphism of an object $𝑋 \in 𝖢$ agrees with the component of $i d$ in $𝑋$ . cat