I am working on formalizing software design using category theory.
However the most natural way for me to express what I want is with a Category where multiple morphisms can join into a single morphism.
Thinking diagrammatically, what I am talking about is multiple arrows (from different objects) merging into a single arrow before reaching an object. Kinda like an inverse arrow split.
Please note that I am not talking about Categories where objects can be (say) concatenations of multiple types. It also does not make sense (for what I am working on) to use currying.
So my question is: has this type of structure been researched? If yes what would be a good reference?