By the universal property of the localization functor $\mathsf{C} \to \cat{Ho}\mathsf{C}$, $\mathbf{L} F$ and $\mathbf{R} F$ are equivalently expressible as homotopical functors $\mathbf{L} F, \mathbf{R} F \colon \mathsf{C} \rightrightarrows \cat{Ho}\mathsf{D}$; these functors are often called the left or right derived functors of $F$. Sometimes, though by no means always, there exists a lift of a left or right derived functor along $\mathsf{D} \to \cat{Ho}\mathsf{D}$. For emphasis, we will call these lifts point-set derived functors, though other authors choose to call them simply ``derived functors’’ \cite{riehl-categorical}.