Name for a "functor" that does not preserve identities.

Question

The definition of a functor $F$ between two categories $\mathcal{C}$ and $\mathcal{D}$ says that:

1. If $f$ is a morphism between objects $A,B \in \mathcal{C}$ then $F(f)$ is a morphism between $F(A),F(B)$.
2. If $f: A \to B$ and $g: B \to C$ then $F(g \circ f) = F(g) \circ F(f)$.
3. For each object $A$, $F(\text{Id}_A) = \text{Id}_{F(A)}$.

Is there a standard name for a mapping that satisfies (1) and (2) but not (3)? In other words, a functor without preserving identities?

Answer 1

It's called a semifunctor (this is the natural notion of morphism between semicategories).