Looking for example at $R$ modules for some commutative $R$, we have the direct sum and the tensor product acting analogously to addition and multiplication. After studying a little bit about co-algebras and bi-algebras, I wondered if there is any way to define something that would be analogous to comultiplicaion in this setting.

What I might imagine is an additive functor $C \to C \otimes C$ when $C$ is an abelian category, under some suitable definition for the tensor (I looked u a little about it and I saw that there are a definitions for that, which I did not fully understand). Also, I can't really imagine what would be an analog for the counit except or maybe (the) functor $C \to *$.

Looking for example at $R$-modules for some commutative $R$, we have the direct sum and the tensor product acting analogously to addition and multiplication. After studying a little bit about co-algebras and bi-algebras, I wondered if there is any way to define something that would be analogous to comultiplication in this setting.

What I might imagine is an additive functor $C \to C \otimes C$ when $C$ is an abelian category, under some suitable definition for the tensor (I looked u a little about it and I saw that there are a definitions for that, which I did not fully understand). Also, I can't really imagine what would be an analog for the counit except or maybe (the) functor $C \to *$.