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To answer the first question provided one has, as you say, (small) products and equalizers the notion of sheaf makes sense as one has the right diagram corresponding to any cover. But we can just say that C $C$ is complete since a category is complete iff it has all products and equalizers.

For the sheafification to exist it is sufficient that C $C$ also be cocomplete so that one can take colimits over suitable categories of covering sieves. This comes up in the construction usually denoted by $(-)^+$ which applied twice to a presheaf results in a sheaf.
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To answer the first question provided one has, as you say, (small) products and equalizers the notion of sheaf makes sense as one has the right diagram corresponding to any cover. But we can just say that C is complete since a category is complete iff it has all products and equalizers.

For the sheafification to exist it is sufficient that C also be cocomplete so that one can take colimits over a suitable category categories of covering sieves. This comes up in the construction usually denoted by (-)^+ which applied twice to a presheaf results in a sheaf.
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To answer the first question provided one has, as you say, (small) products and equalizers the notion of sheaf makes sense as one has the right diagram corresponding to any cover. But we can just say that C is complete since a category is complete iff it has all products and equalizers.

For the sheafification to exist it is sufficient that C also be cocomplete so that one can take colimits over a suitable category of covering sieves. This comes up in the construction usually denoted by (-)^+ which applied twice to a presheaf results in a sheaf.