Is this simple drawing a category?

Question

As far as I can tell this is a category but I am not 100% sure. Objects are A, B, C. Arrows are f and g.

Could someone please confirm that this drawing is indeed a category? So no arrows need to be between C and B ?

Answer 1

(I don't know why the people are using again comments to answer the question; this way even "answered" questions will stay on the list of unanswered questions.)

Yes, this is a category. Actually, every preordered set (a set with a reflexive and transitive relation $\leq$) can be regarded as a category in a natural way, where there is exactly one morphism $x \to y$ when $x \leq y$.

In your case, you consider the preordered set $\{A,B,C\}$ with $A \leq B$ and $A \leq C$ (and of course $A \leq A$, $B \leq B$, $C \leq C$).