Is there a well-known term for a category where, equivalently:
- every morphism is a monomorphism
- every slice category is a preorder
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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.