Equal Categories

Question

Let $\infty$ be the "category" of all categories, where the objects are categories and the morphisms are functors. I am trying to motivate the definition of equal categories by doing it as follows.

We say two categories are equal $C=D$, or isomorphic, if there exist functors $F:C\to D$ and $G:D\to C$ such that $G\circ F = \text{id}_C$ and $F\circ G = \text{id}_D$. By $=$ of functors, we mean equality within the category of functors, i.e. $G\circ F = \text{id}_C$ in $\text{Fun}(C,C)$ and $F\circ G = \text{id}_D$ in $\text{Fun}(D,D)$. Thus, there exist natural transformations $\varphi: G\circ F \to \text{id}_C$ and $\psi: F\circ G\to \text{id}_D$ which are isomorphisms.

How does one use this approach to show that it leads to the standard definition of equal categories? (i.e. $F(\varphi_A) = \psi_{F(A)}$ and $G(\psi_B) = \varphi_{G(B)}$).

Answer 1

I think you are confusing three different notions here.

1. Things are equal when they are equal.
   Notation: $A=B$.
2. Two objects are isomorphic in a category if there exists an invertible arrow between them.
   Notation: $A\cong B$.
3. Two categories ($A$ and $B$) are equivalent if there exists a pair of functors $F,G$ and a pair of natural isomorphisms $G\circ F\to id$ and $F\circ G\to id$.
   Notation: $A\simeq B$.