Suppose $F:\mathcal{X}\rightarrow\mathcal{Y}$ is gerbe over stack and $p:X\rightarrow \mathcal{X}$ is an atlas $\mathcal{X}$.

Does this imply $F\circ p:X\rightarrow \mathcal{Y}$ is an atlas for $\mathcal{Y}$?

Atlas of a stack is as in Understanding the definition of atlas of a stack

Gerbe over a stack is as in Understanding definition of gerbe over a stack