In Category Theory, we call a fork a diagram
$$X\xrightarrow{\;e\;}A\begin{array}{c} \xrightarrow{\;f\;}\\ \xrightarrow[\;g\;]{} \end{array}B$$
that commutes, i.e. $f \circ e = g \circ e$.
In this context, what can we call $e$? Is there a proper name for it? I know it is not necessarily an equalizer of $f$ and $g$.
What would I call a generalization of this concept in the following sense: given objects $A$ and $B$, a non-empty subset $H \subseteq Hom(A,B)$ and a morphism $e : X \rightarrow A$, we have that
$$\forall f,\, g\, \in H.\, f \circ e = g \circ e$$
In particular, I am interested in the case where $H$ is the set of isomorphisms between $A$ and $B$.