n-category = (n,n)-category
n-groupoid = (n,0)-category
The notion of essential image is supposed to be an adaptation of the notion of image from a 1-categorical to the 2-categorical context Cat, i.e. to the image of functors. But some care has to be exercised.
For another definition of image of a functor, see (2,1)-image.
(A concrete realization of) the essential image of a functor between categories or -categories is the smallest replete subcategory of the target -category containing the image of . (The image is, in turn, the smallest subcategory which contains all the -cells which are strictly the images of -cells in .)
Note that if is not pseudomonic, then its essential image, defined in this way, need not be equivalent to its ordinary image.
Note that the property of “belonging to the image” (said of an object or morphism) is evil; of two equivalent objects, one may belong while the other does not. Passing to the essential image removes this, so that the property of “belonging to the essential image” is no longer evil.
Of course, the property of “being equal to the essential image” (said of a subcategory) is evil, as is the property of “being replete”. But is a replete subcategory of if and only if the property of belonging to (said of an object or morphism) is not evil.