Q:

Category where every Morphism is a Mono

Is there a well-known term for a category where, equivalently:

every morphism is a monomorphism
every slice category is a preorder

?

A:

A category in which there is at most one morphism between two objects is called thin. A category for which each slice is a gizmo category is often called a locally guizmo category (for example locally cartesian closed category).
Hence it makes sense to call a category in which every morphism is mono a locally thin category.

Beware though that it is not standard and I wouldn't use it without redefining it first.

