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

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

This question is stated correctly in geometric stacks sense and I am sure this question makes sense in algebraic set up as well. Please feel free to edit to make it suitable for algebraic set up (if you understand what I am intended to ask).

As there are more algebraic geometry’s, I thought it is better to pose question in that form.

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 over the category of manifolds

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