While reading Brylinski I am trying to understand the descent of morphisms of sheaves.

In the "new" definition of a sheaf $A$ over a space $X$, we associate to each surjective local homeomorphism $f:Y \to X$ a set. The "restriction" condition of a presheaf amounts to, given any diagram $$W \xrightarrow{h} Z \xrightarrow{g} Y \xrightarrow{f} X$$ having $(gh)^{-1} = h^{-1} \circ g^{-1}$ as pullbacks $A(Y\xrightarrow{f}X) \to A(W \xrightarrow{fgh} Z)$.

I want to say that if $A$ is already a sheaf then the above property is satisfied. My proof feels trivial, hence my worry. Also, I am uneasy since Brylisnki doesn't state this fact but instead says it "should" be true. He later comments that as functors from the category of sheaves on $Y$ to the category of sheaves on $Y$ these two maps are NOT equal; but there is a natural transformation.

Am I right in feeling that that original "restriction" property is still automatically satisfied despite his cautions about these maps as functors??