Name of this property: all maps from given class of spaces into $X$ are nullhomotopic?

Question

Let $X$ be a topological space and let $\mathcal{C}$ be some class of topological spaces. Is there a standard name for the following property of $X$?

For every space $C\in \mathcal{C}$ all maps $C\to X$ are nullhomotopic.

(Of course if $X\in\mathcal{C}$, then this property is just contractibility.)