### Algebraic Geometry 2: Philosophizing Categories

Faithful functor: Let $F:A\to B$ be a functor between two categories. If the map Mor$(A,A')\to$ Mor$(F(A),F(A'))$ is injective, then the functor is called faithful. For example, the functor between the category of sets with only bijective mappings qualified to be morphisms between objects, to the category of sets with all kinds of mappings allowed to be morphisms, carrying objects and morphisms to themselves, is faithful.
Full functor: If Mor$(A,B)\to$ Mor$(F(A),F(B))$ is surjective, then the functor is called full. For example, the functor defined above is not full.

Natural transformation of covariant functors: Let $F,G$ be functors between categories $A$ and $B$. Let $M,M'\in A$ be objects, and let $f:M\to M'$ be morphisms between them. Then $m:F\to G$ is a natural transformation if the following diagram commutes:

Philosophical point: Why are commutative diagrams so omnipresent and important in Mathematics? It seems to be a fairly arbitrary condition to satisfy! Commuting diagrams essentially signify that “similar things” are happening at “different places”, and the “similar things” can easily be inter-converted. Too hand-wavy? Please bear with me.

Say $M,M',F(M),F(M'),G(M),G(M')$ are all $\Bbb{Z}$, and $m(\Bbb{Z})$ is idenity. Also assume that  $F(f)$ maps $k\to k+1$. Then if $G(f)$ also maps $k\to k+1$, then $m$ is a natural transformation. However, if $G(f)$ is of any other description, then $m$ is not a natural transformation.

Forming other examples should convince you of the fact above.