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

In trying to form a new definition of a presheaf $A$ over a space $X$, we associate to each surjective local homeomorphism $f:Y \to X$ a set, denoted $A(Y\xrightarrow{f}X)$. The "restriction" condition of a presheaf amounts to: given a surjective local homeomorphism $g:Z \to Y$ we have a pullback map $g^{-1}:A(Y\xrightarrow{f}X) \to A(Z \xrightarrow{fg}X)$. The transitivity property for these "restriction" (pullback) maps is that given any diagram $$W \xrightarrow{h} Z \xrightarrow{g} Y \xrightarrow{f} X$$ having $(gh)^{-1} = h^{-1} \circ g^{-1}$ as pullbacks $A(Y\xrightarrow{f}X) \to A(W \xrightarrow{fgh} Z)$. $\\ \\\\$

If $A$ is already a presheaf, in the good 'ol fashioned sense, then we can define our assignment $A(Y\xrightarrow{f}X)$ to be the global sections of $Y$ given by the inverse image of $A$ on $X$, i.e. $\Gamma(Y, f^{-1}A)$

I have 2 questions:

1. Is it true that if $A$ is already a sheaf in the good 'ol fashioned sense, then the above property (transitivity of the "restriction") 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.

2. He later comments that as functors from the category of sheaves on $W$ to the category of sheaves on $Y$ , $h^{-1}\circ g^{-1}$ and $(gh)^{-1}$ are NOT equal; but there is a natural transformation. Why are these two functors not equal? It seems like they send the same sheaves to the same places, unless of course I am making identifications of categories that I don't realize?

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

In trying to form a new definition of a presheaf $A$ over a space $X$, we associate to each surjective local homeomorphism $f:Y \to X$ a set, denoted $A(Y\xrightarrow{f}X)$. The "restriction" condition of a presheaf amounts to: given a surjective local homeomorphism $g:Z \to Y$ we have a pullback map $g^{-1}:A(Y\xrightarrow{f}X) \to A(Z \xrightarrow{fg}X)$. The transitivity property for these "restriction" (pullback) maps is that given any diagram $$W \xrightarrow{h} Z \xrightarrow{g} Y \xrightarrow{f} X$$ having $(gh)^{-1} = h^{-1} \circ g^{-1}$ as pullbacks $A(Y\xrightarrow{f}X) \to A(W \xrightarrow{fgh} Z)$. $\\ \\\\$

If $A$ is already a presheaf, in the good 'ol fashioned sense, then we can define our assignment $A(Y\xrightarrow{f}X)$ to be the global sections of $Y$ given by the inverse image of $A$ on $X$, i.e. $\Gamma(Y, f^{-1}A)$

I have 2 questions:

1. Is it true that if $A$ is already a sheaf in the good 'ol fashioned sense, then the above property (transitivity of the "restriction") 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.

2. He later comments that as functors from the category of sheaves on $Y$ to the category of sheaves on $W$ , $h^{-1}\circ g^{-1}$ and $(gh)^{-1}$ are NOT equal; but there is a natural transformation. Why are these two functors not equal? It seems like they send the same sheaves to the same places, unless of course I am making identifications of categories that I don't realize?

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??

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

In trying to form a new definition of a presheaf $A$ over a space $X$, we associate to each surjective local homeomorphism $f:Y \to X$ a set, denoted $A(Y\xrightarrow{f}X)$. The "restriction" condition of a presheaf amounts to: given a surjective local homeomorphism $g:Z \to Y$ we have a pullback map $g^{-1}:A(Y\xrightarrow{f}X) \to A(Z \xrightarrow{fg}X)$. The transitivity property for these "restriction" (pullback) maps is that given any diagram $$W \xrightarrow{h} Z \xrightarrow{g} Y \xrightarrow{f} X$$ having $(gh)^{-1} = h^{-1} \circ g^{-1}$ as pullbacks $A(Y\xrightarrow{f}X) \to A(W \xrightarrow{fgh} Z)$. $\\ \\\\$

If $A$ is already a presheaf, in the good 'ol fashioned sense, then we can define our assignment $A(Y\xrightarrow{f}X)$ to be the global sections of $Y$ given by the inverse image of $A$ on $X$, i.e. $\Gamma(Y, f^{-1}A)$

I have 2 questions:

1. Is it true that if $A$ is already a sheaf in the good 'ol fashioned sense, then the above property (transitivity of the "restriction") 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.

2. He later comments that as functors from the category of sheaves on $W$ to the category of sheaves on $Y$ , $h^{-1}\circ g^{-1}$ and $(gh)^{-1}$ are NOT equal; but there is a natural transformation. Why are these two functors not equal? It seems like they send the same sheaves to the same places, unless of course I am making identifications of categories that I don't realize?