This is Theorem 6.1.3.9 (3) in HTT.

]]>I think a special case of this (for sheaves over spaces) is given as Exercise 2.7.D of Vakil’s *Rising Sea*. See also Remark 2.7.5.

Consider the presheaf Sh on S valued in the bicategory of categories

that sends s∈S to the category of sheaves over the slice site S/s.

One can show that Sh is actually a sheaf of categories (or rather a stack in bicategories, to be precise).

In other words, a sheaf can be specified locally on the elements of some cover and then glued together,

and the same is true for morphisms of sheaves.

Is this fact somewhere in the literature?

I am also interested in the versions for ∞-sheaves on ∞-sites (any model will do). ]]>