TITLE: Is Category Theory similar to Graph Theory?
QUESTION [34 upvotes]: The following author noted:

Roughly speaking, category theory is graph theory with additional structure to represent composition.

My question is: Is Category Theory similar to Graph Theory?

REPLY [2 votes]: A deductive system is a graph in which to each object $A$ there is associated an arrow $1_A: A\to A$, the identity arrow, and to each pair of arrows $f:A\to B$ and $g:B\to C$ there is associated an arrow $gf:A\to C$, the composition of $f$ and $g$.
A logician may think of the objects as formulas and of the arrows as deductions or proofs, hence of
$$\frac{f:A\to B \qquad g:B\to C}{gf:A\to C}$$
as a rule of inference.
A category is a deductive system in which the following equations hold for all $f:A\to B$, $g:B\to C$ and $h:C\to D$:
$$f1_A=f=1_Bf$$
$$(hg)f=h(gf)$$

(Lambek & Scott. Introduction to higher order categorical logic, p. 5)

In summary: Lambek and Scott define a deductive system as a (directed) graph with units and compositions (alias modus ponens, alias rule of inference) and a category as a deductive system where the 'ususal' laws hold for units and rule of inference.