While reading [Brylinski][1]  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??**


  [1]: http://books.google.com/books?id=ta5UB1D64_gC&printsec=frontcover&dq=brylinski+loop+spaces&hl=en&sa=X&ei=jtmqT724Nurl6QGF55SxBA&ved=0CDMQ6AEwAA#v=onepage&q=brylinski%2520loop%2520spaces&f=false