Given a category \(\mathcal{C}\), show that there exists a functor \(id_\mathcal{C}\) known as the identity functor on \(\mathcal{C}\)
Show that given \(\mathcal{C}\xrightarrow{F}\mathcal{D}\) and \(\mathcal{D}\xrightarrow{G}\mathcal{E}\) we can define a new functor \(\mathcal{C}\xrightarrow{F;G}\mathcal{E}\) just by composing functions.
Show that there is a category, let’s call it Cat where the objects are categories, morphisms are functors, and identities/composition are given as above.
Mapping objects and morphisms to themselves satsifies the functor constraints of preserving identities and composition.
If \(F,G\) both independently preserve identity arrows, then composition of these will also preserve this. Also \(G(F(f;g))=G(F(f);F(g))=G(F(f));G(F(g))\) using the independent facts that \(F,G\) each preserve composition.
Composition of identity functions do not change anything, so the identity functor (defined by identity function) will obey unitality. Because function composition is associative and functor composition is defined by this, we also satisfy that constraint.