As Fernando points out, you can't talk about isomorphisms in a semicategory, which means that they won't be as much use as categories in describing universes of mathematical objects. But the category of semicategories has a surprisingly interesting relationship to that of categories. There is of course a forgetful functor $\mathrm{Semicat} \mathrm{Cat} \to \mathrm{Cat}$, mathrm{Semicat}$, and as Scott says it has a left adjoint that does what you expect. But it also has a right adjoint, which takes a semicategory S to the category of idempotents in S: the objects are idempotents$e \colon a \to a$and a morphism$e \to e'$is a morphism$f \colon a \to a'$in S such that$fe = f = e'f$. So we get a monad on Cat whose unit is the canonical functor from a category to its idempotent-splitting completion, or Cauchy completion, or Karoubi envelope. Böhm, Lack and Street use this framework here to talk about weak Hopf algebras. They show that 'weak monoids' fall naturally out of the formal theory of monads if instead of working directly in a bicategory you Cauchy-complete the hom-categories first. Another application of semicategories and semifunctors is in computer science: Hayashi, Adjunction of semifunctors: categorical structures in nonextensional$\lambda$-calculus, TCS 41, shows how to describe$\lambda$-calculus without the$\eta$-law quite elegantly. I haven't worked it out, but it seems to me that this framework should also give a way of talking about 'weak limits' (the kind with not-necessarily-unique mediating morphisms) in terms of adjunctions. 1 As Fernando points out, you can't talk about isomorphisms in a semicategory, which means that they won't be as much use as categories in describing universes of mathematical objects. But the category of semicategories has a surprisingly interesting relationship to that of categories. There is of course a forgetful functor$\mathrm{Semicat} \to \mathrm{Cat}$, and as Scott says it has a left adjoint that does what you expect. But it also has a right adjoint, which takes a semicategory S to the category of idempotents in S: the objects are idempotents$e \colon a \to a$and a morphism$e \to e'$is a morphism$f \colon a \to a'$in S such that$fe = f = e'f$. So we get a monad on Cat whose unit is the canonical functor from a category to its idempotent-splitting completion, or Cauchy completion, or Karoubi envelope. Böhm, Lack and Street use this framework here to talk about weak Hopf algebras. They show that 'weak monoids' fall naturally out of the formal theory of monads if instead of working directly in a bicategory you Cauchy-complete the hom-categories first. Another application of semicategories and semifunctors is in computer science: Hayashi, Adjunction of semifunctors: categorical structures in nonextensional$\lambda$-calculus, TCS 41, shows how to describe$\lambda$-calculus without the$\eta\$-law quite elegantly. I haven't worked it out, but it seems to me that this framework should also give a way of talking about 'weak limits' (the kind with not-necessarily-unique mediating morphisms) in terms of adjunctions.