For a monoidal category $(C,\otimes)$, and $X$ an object of $C$, we can define an endofunctor on $C$ which acts on objects as $$ X \otimes -: Y \mapsto X \otimes Y, $$ and on morphisms as $$ \big[f: Y \to Z\big] \mapsto \big[\text{id} \otimes f: X \otimes Y \to X \otimes Z\big]. $$
Do such functors have a name in general?