Are categories just multi-digraphs with extra restrictions?

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From my (limited) understanding, a category is just a class of objects (which form the "vertices") and a class of morphisms between those objects, plus the requirement that associativity and identity hold. Whenever I see them drawn, all I see are multi-digraphs. This being the case, it seems that "category theory" boils down to graph theory. I'm assuming this isn't actually true. What can be expressed by category theory that can't be expressed just by interpreting a multi-digraph in a particular way?