Definition 2.2.1. A presheaf (resp. copresheaf) on a metric site C with values in a category D is a contravariant (resp. covariant) functor A: C^op ® D (resp. C ® D) from C to D. We say A is a sheaf (resp. cosheaf) if for any object X of C, and any open effective cover {U_i} of X, A(X) is the limit (resp. colimit) of the system A(U_ij) for all i, j, where U_ij = U_i Ç U_j Í |X|, via the morphisms induced by the inclusions among U_ij. 

Denote by [C, D] (resp. {C, D}) the category of copresheaves (resp. cosheaves) on C with values in D. We denote by [C_D] the subcategory of [C, D] consisting of functors preserve colimits. If colimits exist D, then colimits exist in [C, D] which are computed pointwise. 

Suppose E is a category. Any functor j: C ® E determines a functor j*: [E, D] ® [C, D] sending each copresheaf B Î [C, D] to j*(A) = Aj Î [E, D]. Suppose colimits exist in D. Then j* has a left adjoint j!: [C, D] ® [E, D]. If A Î [C, D] we say that j!(A) is the Kan extension of A (along j). 

If D is the category of sets (resp. abelian groups, resp. rings), a presheaf or sheaf on C with values in D is called a presheaf (or sheaf) of sets (resp. abelian groups, resp. rings) on C. 

Remark 2.2.2. Suppose C is a metric site and A is a presheaf (resp. copresheaf) on C with values in a category D. Then A is a sheaf (resp. cosheaf) if and only if for any Z Î D, the presheaf of sets on C given by X ® hom (Z, A(X)) (resp. hom (A(X), Z)) is a sheaf. 

Remark 2.2.3. A presheaf A of sets on a metric site C is a sheaf if and only if the following condition is satisfied:  
If X is an object, then to give an element f of A(X), it is equivalent to give an open effective cover {U_i} of |X|, together with f_i Î A(U_i), such that the restrictions of f_i and f_j to U_i Ç U_j are the same.  

Denote by C^{^} and C^~ the categories of presheaves and sheaves of sets on a metric site C respectively. We may identify C with a subcategory of C^{^} via Yoneda embedding; the inclusion functor C ® C^{^} preserves limits.  

Remark 2.2.4. (a) Suppose j: C ® D is a functor. Then (2.2.2) implies that j is a cosheaf if and only if j*(D) Í C^~.  
(b) If C is strict then C Í C^~. 

Example 2.2.5. (a) A metric site C is strict if and only if the identity functor C ® C is a cosheaf (this follows from (2.2.2) and (2.2.3)).  
(b) The functor t: C ® Top is a cosheaf for any metric site C. 

Proposition 2.2.6. If colimits exist in D, then the same is true for {C, D}. Colimits in {C, D} are computed pointwise. The inclusion functor {C, D} ® [C, D] preserves colimits. 

Proof. It suffices to consider the case that C = W(X) for a topological space X. The assertions then follow from the transitivity of colimits in [W(X), D]. 

Example 2.2.7. We have C^~ = {C, Set^op}^op. Since Set^op has colimits, {C, Set^op} has colimits and the inclusion {C, Set^op} ® [C, Set^op] preserves colimits. It follows that limits exist in C^~ and the inclusion C^~ ® C^{^} preserves limits.  

Remark 2.2.8. (2.2.7) also follow from that fact that C^~ is an isometric reflective subcategory of C^{^} with the reflector (+): C^{^} ® C^~: a presheaf O Î C^{^} induces a presheaf O_X on the space |X| of any object X, generated by O(U) for effective open sets U of |X|. Denote by O_X⁺ the associated sheaf to O_X. Then X ® O_X⁺(|X|) defines a sheaf O⁺ of sets on C, which is the associated sheaf to O. It is easy to see that (+) preserves finite limits.  

Proposition 2.2.9. Suppose C' is a metric site and C a full subsite of C'. Suppose D is a category with colimits and j: C ® D is a cosheaf. Then the Kan extension j' of j on C' is a cosheaf.  

Proof. Suppose X Î C'. Then j' induces a copresheaf j'|_X on |X|. j'|_X is the colimit of the direct images of j'|_Y = j|_Y on |Y| for all the morphisms f: Y ® X with Y Î C. Since each j|_Y is a cosheaf, j'|_X is a cosheaf by (2.2.6). Thus j': C' ® D is a cosheaf.  

Remark 2.2.10. If C is a Cauchy-base (resp. finite Cauchy-base, resp. effective-base) of C', for any cosheaf j: C d D, the Kan extension j': C' d D is called the cosheaf on C' generated by j. The correspondence j d j' is an equivalence from {C, D} to {C', D}. 

2.2.11 The Yoneda embedding Y: C ® C^{^} is a universal copresheaf with values in a category with colimits in the following sense:  
Suppose D is a category with colimits. For any presheaf j: C ® D the Kan extension j^{^}: C^{^} ® D preserves colimits because j^{^} is the left adjoint functor of the restriction of j*: D^{^} ® C^{^} on D (2.2.4a). Conversely any colimits preserving functor from C^{^} d D arises in this way since any presheaf in C^{^} is a colimit of representable presheaves. Thus we obtain an equivalence (^): [C, D] ® [C^{^}_D].  

2.2.12 Suppose C is strict. The embedding S: C ® C^~ is a cosheaf. It is a universal cosheaf on C with values in a category with colimits in the following sense:  
Suppose D is a category with colimits. For any cosheaf j: C ® D we denote by j^~ the Kan extension of j on C^~ along S. Then j^~: C^~ ® D is a left adjoint of the functor j*|_D: D ® C^~; thus j^{^} preserves colimits. Conversely any colimits preserving functor from C^~ ® D arises in this way since any A Î C^~ is a colimit of representable presheaves and A|_C is a cosheaf. Thus we obtain an equivalence (~): {C, D} ® [C^~_D].