Canonical morphism from $\operatorname{ran}_{(\operatorname{lan}_{f} 1_X)} 1_B$ to $f: X \to B$

Question

For a given morphism $f: X \to B$, and assuming the extensions exist, does the canonical 2-morphism $ \operatorname{ran}_{(\operatorname{lan}_{f} 1_X)} 1_B \Rightarrow f$ has a conventional name ?

The same question could be asked for $ f \to \operatorname{lift}_{(\operatorname{rift}_f 1_B)} 1_X$