Addition modulo 3 can't be a morphism on a three object category, because addition is a binary operation. It has two inputs and one output. A morphism has one input and one output.
You could take the objects 1,2,3 together with the morphism (plus 1) mod 3. In which case the resulting category is simply a triangle.
A finite category, is of course, just a multigraph with loops.
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Addition modulo 3 can't be a morphism on a three object category, because addition is a binary operation. It has two inputs and one output. A morphism has one input and one output.
You could take the objects 1,2,3 together with the morphism (plus 1) mod 3. In which case the resulting category is simply a triangle.
A finite category, is of course, just a multigraph with loops.