Name for a certain kind of "factor object" in a category

Question

Suppose that $\mathcal{C}$ is a category and $B$ is an object of $\mathcal{C}$ with the property that there is another object $C$ different from $B$, such that every morphism $f_1:A \to B$ factors through $C$, that is there are morphisms $f_2:A \to C$ and $f_3: C\to B$, such that $f_1=f_3\circ f_2$.

Is there a special name for object $C$? (I have no particular example in mind)

Answer 1

$C$ is an object that retracts onto $B$. The identity of $B$ factors through $C$, so indeed there exists a retraction $i : B \to C$ and $r : C \to B$ such that $r \circ i = id_B$. Conversely, given such a retraction, if $f_1 : A \to B$ is any morphism, let $f_2 = i \circ f$ and $f_3 = r$.

I don't really know a name for $C$. Usually we say that $B$ is a retract of $C$.