Here is an elementary example of a ubiquitous category (a monoid) in which every morphism is a bi-morphism, but the only isomorphism is the identity morphism.

Fix a set $X$.

Consider the set of all finite sequences $x_1$,$x_2$,..$x_n$ such that each $x_i \in X$. We also allow the empty sequence.

Under cancatantion, the collection of all such sequences form the morphisms of a category with one object, such that each morphism is a bimorphism, and such that the only isomorphism is the identity morphism.

Every strict poset is naturally a thin category (in which there is at most one morphism between objects), and in particular all nontrivial morphisms are both monic and epic (and hence bimorphisms), but not isomorphisms.

Here is an elementary example of a ubiquitous category (a monoid) in which every morphism is a bi-morphism, but the only isomorphism is the identity morphism.

Fix a set $X$.

Consider the set of all finite sequences $x_1$,$x_2$,..$x_n$ such that each $x_i \in X$. We also allow the empty sequence.

Under cancatantion, the collection of all such sequences form the morphisms of a category with one object, such that each morphism is a bimorphism, and such that the only isomorphism is the identity morphism.

Every strict poset is naturally a thin category (in which there is at most one morphism between objects), and in particular all nontrivial morphisms are both monic and epic (and hence bimorphisms), but not isomorphisms.

The category whose morphisms are based covers between connected n-manifolds is not thin, but every morphism is both monic and epic, and often not an isomorphism.

Here is an elementary example of a ubiquitous category (a monoid) in which every morphism is a bi-morphism, but the only isomorphism is the identity morphism.

Fix a set $X$.

Consider the set of all finite sequences $x_1$,$x_2$,..$x_n$ such that each $x_i \in X$. We also allow the empty sequence.

Under cancatantion, the collection of all such sequences form the morphisms of a category with one object, such that each morphism is a bimorphism, and such that the only isomorphism is the identity morphism.

Here is an elementary example of a ubiquitous category (a monoid) in which every morphism is a bi-morphism, but the only isomorphism is the identity morphism.

Fix a set $X$.

Consider the set of all finite sequences $x_1$,$x_2$,..$x_n$ such that each $x_i \in X$. We also allow the empty sequence.

Under cancatantion, the collection of all such sequences form the morphisms of a category with one object, such that each morphism is a bimorphism, and such that the only isomorphism is the identity morphism.

Every strict poset is naturally a thin category (in which there is at most one morphism between objects), and in particular all nontrivial morphisms are both monic and epic (and hence bimorphisms), but not isomorphisms.