Are there important or interesting (nontrivial) examples of categories where the objects and morphisms are the same structures (e.g. all sets, or all functions)?
For instance, consider the construct $\mathbf{C}$ whose objects are functions and morphisms are functions, defined by $\hom_\mathbf{C}(f, g) = \hom_\mathbf{Set}(\operatorname{dom}(f), \operatorname{dom}(g))$. The forgetful functor $U : \mathbf{C} \to \mathbf{Set}$ that maps objects to their domain and morphisms to themselves.
This is a rather trivial and uninteresting example. Are there more interesting examples in research?