Identity Functor is Left Identity

Theorem
Let $\mathbf C$ and $\mathbf D$ be metacategories.

Let $F: \mathbf C \to \mathbf D$ be a functor, and let $1_{\mathbf D}$ be the identity functor on $\mathbf D$.

Then the composite functor $1_{\mathbf D} F$ satisfies:


 * $1_{\mathbf D} F = F$

Proof
We have, for all objects $C$ of $\mathbf C$:


 * $1_{\mathbf D}F C = 1_{\mathbf D} \left({F C}\right) = F C$

by definition of composition of functors and of identity functor.

Similarly, we have, for a morphism $f$ of $\mathbf C$:


 * $1_{\mathbf D}F f = 1_{\mathbf D} \left({F f}\right) = F f$

Hence the result.